Pfaffian Schur Processes and Last Passage Percolation in a Half-Quadrant

PFAFFIAN SCHUR PROCESSES AND LAST PASSAGE PERCOLATION IN A HALF-QUADRANT

JINHO BAIK, GUILLAUME BARRAQUAND, IVAN CORWIN, AND TOUFIC SUIDAN

Abstract. We study last passage percolation in a half-quadrant, which we analyze within the framework of Pfaffian Schur processes. For the model with exponential weights, we prove that the fluctuations of the last passage time to a point on the diagonal are either GSE Tracy-Widom distributed, GOE Tracy-Widom distributed, or Gaussian, depending on the size of weights along the diagonal. Away from the diagonal, the fluctuations of passage times follow the GUE Tracy- Widom distribution. We also obtain a two-dimensional crossover between the GUE, GOE and GSE distribution by studying the multipoint distribution of last passage times close to the diagonal when the size of the diagonal weights is simultaneously scaled close to the critical point. We expect that this crossover arises universally in KPZ growth models in half-space. Along the way, we introduce a method to deal with diverging correlation kernels of point processes where points collide in the scaling limit.

Contents 1. Introduction 1 2. Fredholm Pfaffian formulas8 3. Pfaffian Schur process 13 4. Fredholm Pfaffian formulas for k-point distributions 23 5. Convergence to the GSE Tracy-Widom distribution 33 6. Asymptotic analysis in other regimes: GUE, GOE, crossovers 46 References 56

1. Introduction 2 In this paper, we study last passage percolation in a half-quadrant of Z . We extend all known results for the case of geometric weights (see discussion of previous results below) to the case of exponential weights. We study in a unified framework the distribution of passage times on and off diagonal, and for arbitrary boundary condition. We do so by realizing the joint distribution of last passage percolation times as a marginal of Pfaffian Schur processes. This allows us to use powerful methods of Pfaffian point processes to prove limit theorems. Along the way, we discuss an issue arising when a simple point process converges to a point process where every point has multiplicity two, which we expect to have independent interest. These results also have consequences about interacting particle systems – in particular the TASEP on positive integers and the facilitated TASEP – that we discuss in [2]. Definition 1.1 (Half-space exponential weight LPP). Let wn,m be a sequence of i.i.d. n>m>0 1 exponential random variables with rate 1 when n > m + 1 and with rate α when n = m. We

1 The exponential distribution with rate α ∈ (0, +∞), denoted E(α), is the probability distribution on R>0 such −αx that if X ∼ E(α), P(X > x) = e . 1 m

w22

w11 w21 w31 n

Figure 1. LPP on the half-quadrant. One admissible path from (1, 1) to (n, m) is shown in dark gray. H(n, m) is the maximum over such paths of the sum of the weights wij along the path. deﬁne the exponential last passage percolation (LPP) time on the half-quadrant, denoted H(n, m), by the recurrence for n > m, ( n o max H(n − 1, m),H(n, m − 1) if n > m + 1, H(n, m) = wn,m + H(n, m − 1) if n = m with the boundary condition H(n, 0) = 0.

Remark 1.2. It is useful to notice that the model is equivalent to a model of last passage percolation on the full quadrant where the weights are symmetric with respect to the diagonal (wij = wji). Remark 1.3. If one imagines that the light gray area in Figure1 corresponds to the percolation cluster at some time, its border (shown in black in Figure1) evolves as the height function in the Totally Asymmetric Simple Exclusion process on the positive integers with open boundary condition, so that all our results could be equivalently phrased in terms of the latter model.

1.1. Previous results in (half-space) LPP. The history of fluctuation results in last passage percolation goes back to the solution to Ulam’s problem about the asymptotic distribution of the size of the longest increasing subsequence in a uniformly random permutation [4]. A proof of the nature of fluctuations in exponential distribution LPP is provided in [29]. On a half-space, [5,6,7] studies the longest increasing subsequence in random involutions. This is equivalent to a half-space LPP problem since for an involution σ ∈ Sn, the graph of i 7→ σ(i) is symmetric with respect to the first diagonal. Actually, [5,6,7] treat both the longest increasing subsequences of a random involution and symmetrized LPP with geometric weights. That work contains the geometric weight half-space LPP analogue of Theorem 1.4. The methods used therein only worked when restricted to the one point distribution of passage times exactly along the diagonal. Further work on half-space LPP was undertaken in [38], where the results are stated there in terms of the discrete polynuclear growth (PNG) model rather than the equivalent half-space LPP with geometric weights. The framework of [38] allows to study the correlation kernel corresponding to multiple points in the space direction, with or without nucleations at the origin. In the large scale limit, the authors performed a non-rigorous critical point derivation of the limiting correlation kernel in certain regimes and found Airy-type process and various limiting crossover kernels near the origin, much in the same spirit as our results (see the discussion about these results in Section 1.3). 2 1.2. Previous methods. A key property in the solution of Ulam’s problem in [4] was the relation to the RSK algorithm which reveals a beautiful algebraic structure that can be generalized using the formalism of Schur measures [33] and Schur processes [34]. RSK is just one of a variety of Markov dynamics on sequences of partitions which preserves the Schur process [17, 13]. Studying these dynamics gives rise to a number of interesting probabilistic models related to Schur processes. In the early works of [4,6] the convergence to the Tracy-Widom distributions was proved by Riemann- Hilbert problem asymptotic analysis. Subsequently Fredholm determinants became the preferred vehicle for asymptotic analysis. In a half-space, [5] applied RSK to symmetric matrices. Subsequently, [37] (see also [26]) showed that geometric weight half-space LPP is a Pfaffian point process (see Section 4.1) and computed the correlation kernel. Later, [21] defined Pfaffian Schur processes generalizing the probability measures defined in [37] and [38]. The Pfaffian Schur process is an analogue of the Schur process for symmetrized problems. 1.3. Our methods and new results. We consider Markov dynamics similar to those of [17] and show that they preserve Pfaffian Schur processes. This enables us to show how half-space LPP with geometric weights is a marginal of the Pfaffian Schur process. We then apply a general formula giving the correlation kernel of Pfaffian Schur processes [21] to study the model’s multipoint distributions. We degenerate our model and formulas to the case of exponential weight half-space LPP (previous works [5,6,7, 38] only considered geometric weights). We perform an asymptotic analysis of the Fredholm Pfaffians characterizing the distribution of passage times in this model, which leads to Theorems 1.4 and 1.5. We also study the k-point distribution of passage times at a distance O(n2/3) away from the diagonal, and introduce a new two-parameter family of crossover distributions when the rate of the weights on the diagonal are simultaneously scaled close to their critical value. This generalizes the crossover distributions found in [25, 38]. The asymptotics of fluctuations away from the diagonal were already studied in [38] for the PNG model on a half-space (equivalently the geometric weight half-space LPP). The asymptotic analysis there was non-rigorous, at the level of studying the behavior of integrals around their critical points without controlling the decay of tails of integrals. The proof of the first part of Theorem 1.4 (in Section5) required a new idea which we believe is the most novel technical contribution of this paper: When α > 1/2, H(n, n) is the maximum of a simple Pfaffian point process which converges as n → ∞ to a point process where all points have multiplicity 2. The limit of the correlation kernel does not exist2 as a function (it does have a formal limit involving the derivative of the Dirac delta function). Nevertheless, a careful reordering and non-trivial cancellation of terms in the Fredholm Pfaffian expansions allows us to study the law of the maximum of this limiting point process, and ultimately find that it corresponds to the GSE Tracy-Widom distribution. We actually provide a general scheme for how this works for random matrix type kernels in which simple Pfaffian point processes limit to doubled ones (see Section 5.1). Theorem 5.3 of [38] derives a formula for certain crossover distributions in half-space geometric LPP. This crossover distribution, introduced in [25], depends on a parameter τ (which is denoted η in our Theorem 1.9) and it corresponds to a natural transition ensemble between GSE when τ → 0 and GUE when τ → +∞. Although the collision of eigenvalues should occur in the GSE limit of this crossover, this point was left unaddressed in previous literature and the convergence of the crossover kernel to the GSE kernel was not proved in [25] (see our Proposition 6.12). The formulas for limiting kernels, given by [25, (4.41), (4.44), (4.47)] or [38, (5.37)-(5.40)], make sense so long as τ > 0. Notice a difference of integration domain between [25, (4.44)] and [38, (5.39)]. Our Theorem 1.9 is the analogue of [38, Theorem 5.3] for half-space exponential LPP and we recover

2For a non-simple point process, the correlation functions – Radon-Nikodym derivatives of the factorial moment measures – generically do not exist (see Deﬁnitions in Section 4.1). 3 3 the exact same kernel. When τ = 0 the formula for I4 in [38, (5.39)] involves divergent integrals , though if one uses the formal identity R Ai(x + r)Ai(y + r)dr = δ(x − y) then it is possible to R rewrite the kernel in terms of an expression involving the derivative of the Dirac delta function, and the expression formally matches with the kernel K∞ introduced in Section 5.1. From that (formal) kernel it is non-trivial to match the Fredholm Pfaﬃan with a known formula for the GSE distribution, this matching does not seem to have been made previously in the literature and we explain it in Section 5.1. In random matrix theory, a similar phenomenon of collisions of eigenvalues occurs in the limit from the discrete symplectic ensemble to the GSE [22]. However, [22] used averages of characteristic polynomials to characterize the point process, and computed the correlation kernels from the characteristic polynomials only after the limit, so that collisions of eigenvalues were not an issue. A random matrix ensemble which crossovers between GSE and GUE was studied in [25].

1.4. Other models related to KPZ growth in a half-space. A positive temperature analogue of LPP – directed random polymers – has also been studied. The log-gamma polymer in a half- space (which converges to half-space LPP with exponential weights in the zero temperature limit) is considered in [36] where an exact formula is derived for the distribution of the partition function in terms of Whittaker functions. Some of these formulas are not yet proved and presently there has not been a successful asymptotic analysis preformed of them. Within physics, [28] and [14] studies the continuum directed random polymer (equivalently the KPZ equation) in a half-space with a repulsive and refecting (respectively) barrier and derives GSE Tracy-Widom limiting statistics. Both works are mathematically non-rigorous due to the ill-posed moment problem and certain unproved forms of the conjectured Bethe ansatz completeness of the half-space delta Bose gas. On the side of particle systems, half-space ASEP has first been studied in [41], but the resulting formulas were not amenable to asymptotic analysis. More recently, [9] derives exact Pfaffian formulas for the half-space stochastic six-vertex model, and proves GOE asymptotics for the current at the origin in half-space ASEP, for a certain boundary condition. The exact formulas involve the correlation kernel of the Pfaffian Schur process and the asymptotic analysis in [9] uses some of the ideas developed here.

1.5. Main results. We now state the limit theorems that constitute the main results of this paper. They involve the GOE, GSE and GUE Tracy-Widom distributions, respectively characterized by the distribution functions FGOE, FGSE and FGUE given in Section2 and the standard Gaussian distribution function denoted by G(x). Theorem 1.4. The last passage time on the diagonal H(n, n) satisﬁes the following limit theorems, depending on the rate α of the weights on the diagonal. (1) For α > 1/2, H(n, n) − 4n lim P < x = FGSE (x) . n→∞ 24/3n1/3 (2) For α = 1/2, H(n, n) − 4n lim P < x = FGOE (x) . n→∞ 24/3n1/3 (3) For α < 1/2, n ! H(n, n) − α(1−α) lim P < x = G(x), n→∞ σn1/2

3The saddle-point analysis in the proof of Theorem 5.3 in [38] is valid only when τ > 0. Indeed, [38, (5.44)] requires τ1 + τ2 > η1 + η2 and one needs that η1, η2 > 0 as in the proof of [38, Theorem 4.2]. 4 (1−2α)1/2 where σ = α(1−α) .

This extends the results of [5,6] on the model with geometric weights. The first part of Theorem 1.4 is proved in Section5, while the second and third parts are proved in Section6. Section 6.1 includes an explanation for why the transition occurs at α = 1/2, using a property of the Pfaffian Schur measure (Prop. 3.4). Far away from the diagonal, the limit theorem satisfied by H(n, m) coincides with that of the (unsymmetrized) full-quadrant model. √ √κ Theorem 1.5. For any κ ∈ (0, 1) and α > 1+ κ , we have that √ H(n, κn) − (1 + κ)2n lim P < x = FGUE(x), n→∞ σn1/3 √ (1+ κ)4/3 where σ = √ . κ1/3 This extends the results of [38] on the geometric model to the exponential case and to allow any boundary parameter. Theorem 1.5 is proved in Section 6.3. Remark 1.6. Our asymptotic analyses could be extended to show that the fluctuations of H(n, κn) √ √ for κ ∈ (0, 1) follow the BBP transition [3] when α = κ/(1+ κ)+O(n−1/3), and in particular are 2 √ √ distributed according to FGOE when α = κ/(1 + κ). The BBP transition would also occur if one varies the rate of exponential weights in the first rows of the lattice. Regarding H(n, n), it is not clear if the higher order phase transitions would coincide with the spiked GSE [12, 42]. Our results rely on an asymptotic analysis of the following formula for the joint distribution of passage times. It involves the Fredholm Pfaffian (see Section2 for background on Fredholm Pfaf- fians) of a matrix-valued correlation kernel Kexp defined in Section 4.4 where the next proposition is proved.

Proposition 1.7. For any h1, . . . , hk > 0 and integers 0 < n1 < n2 < ··· < nk and m1 > m2 > ··· > mk such that ni > mi for all i, we have that k ! \ exp P H(ni, mi) < hi = Pf J − K 2 , L Dk(h1,...,hk) i=1 where the R.H.S is a Fredholm Pfaﬃan (Deﬁnition 2.3) on the domain

Dk(h1, . . . , hk) = {(i, x) ∈ {1, . . . , k} × R : x > hi}.

In the one point case k = 1, and for n1 = m1, one could alternatively characterize the probability distribution of H(n, n) by taking the exponential limit of the formulas in [5,6] using for instance the results from [1]. We can generalize the results of Theorems 1.4 and 1.5 by allowing the parameters κ and α to vary in regions of size n−1/3 around the critical values α = 1/2, κ = 1. In the limit, we obtain a new two-parametric family of probability distributions. More generally, we will compute the ﬁnite dimensional marginals of Airy-like processes in various ranges of α and κ. KPZ scaling predictions (together with Theorem 1.5) suggests to deﬁne Hn + n2/3ξη, n − n2/3ξη − 4n + n1/3ξ2η2 H (η) = , n σn1/3 5 GSE GUE $ α = +∞ 1+22/3$k−1/3 α = 2 GSE n = k + 22/3k2/3η 2/3 2/3 m = k − 2 k η Lcross(·; $, η) GUE α = 1/2 GOE GUE η GOE2 Gaussian α = 0 n = 1 n = +∞ m m GUE

Gaussian GOE2

Figure 2. Phase diagram of the fluctuations of H(n, m) as n → ∞ when α and the ratio n/m varies. The gray area corresponds to a region of the parameter space where the fluctuations are on the scale n1/2 and Gaussian. The bounding curve (where fluctuations are expected to be Tracy-Widom GOE2 cf Remark 1.6) asymptotes to zero as n/m goes to +∞. The crossover distribution Lcross(·; $, η) is defined in Definition 2.9 and describes the fluctuations in the vicinity of n/m = 1 and α = 1/2.

4/3 2/3 where η > 0, σ = 2 and ξ = 2 . Let us scale α as 1 + 2σ−1$n−1/3 α = 2 where $ ∈ R is a free parameter. The limiting joint distribution of multiple points is characterized by a new crossover kernel Kcross that we introduce in Section 2.5.

1+22/3$n−1/3 Theorem 1.8. For 0 6 η1 < ··· < ηk, $ ∈ R, setting α = 2 , we have that k ! \ cross lim Hn(ηi) < xi = Pf J − K 2 . n→∞ P L (Dk(x1,...,xk)) i=1

The proof of this result is very similar with the proof of Theorem 1.5. We provide the details of the computations in [2]. In the one-point case (k = 1), this yields a new probability distribution Lcross, depending on two parameters $, η, (see Definition 2.9), which realizes a two-dimensional crossover between GSE, GUE and GOE distributions (see Figure2 and details in Section 2.5). When η = 0, we recover the probability distribution F (x; w) from [6, Definition 4]. When $ = 0, the correlation kernel Kcross degenerates to the orthogonal-unitary transition kernel obtained in [25] and Fcross(x; 0, η) realizes a crossover between FGOE for η = 0 and FGUE for η → +∞. An analogous result was obtained in [38, Theorem 4.2] for the half-space PNG model. In the case when α > 1/2 is fixed, the joint distribution of passage-times is governed by the so-called symplectic-unitary transition [25].

Theorem 1.9. For α > 1/2 and 0 < η1 < ··· < ηk, we have that SU lim (Hn(η1) < x1,...,Hn(ηk) < xk) = Pf J − K 2 , n→∞ P L (Dk(x1,...,xk)) 6 where the kernel KSU is deﬁned in Section 2.5. Theorem 1.9 can be seen as a $ → +∞ degeneration of Theorem 1.8 (see Section 2.5). The proof is very similar with that of Theorem 1.8. We provide the details of the computations in [2]. An analogous result was found for the half-space PNG model without nucleations at the origin [38] although the statement of [38, Theorem 5.3] makes rigorous sense only when τ > 0. The correlation kernel corresponding to the symplectic-unitary transition was studied ﬁrst in [25] 4. This random matrix ensemble is the point process corresponding to the eigenvalues of a Hermitian complex matrix Xη for η ∈ (0, +∞) with density proportional to

−η 2 −Tr (Xη−e X0) exp 1−e−2η , where X0 is a GSE matrix. Various limits of the symplectic-unitary and orthogonal-unitary transition kernels are considered in [25, Section 5]. In particular [25, Section 5.6] considers the limit as η → 0. While the GOE distribution is recovered in the orthogonal-unitary case, the explanations are missing in the symplectic-unitary case (the scaling argument at the end of [25, Section 5.6] is not the reason why one does not recover KGSE). We provide a rigorous proof in Section 6.5. Remark 1.10. One can also study the limiting n-point distribution of √ H(n + n2/3ξη, κn − n2/3ξη) − (1 + κ)2n − n1/3x2 η 7→ , 24/3n1/3 for a fixed κ ∈ (0, 1) and several values of x. In the n → ∞ limit, one would obtain the extended exp exp exp Airy kernel for K12 and 0 for K11 and K22 but we do not pursue that direction. Outline of the paper. In Section2, we provide convenient Fredholm Pfaffian formulas for the Tracy-Widom distributions and their generalizations. In Section3, we define Pfaffian Schur processes and construct dynamics preserving them, thus making a connection to half-space LPP (Proposition 3.10). In Section4, we apply a general result of Borodin and Rains giving the correlation structure of Pfaffian Schur processes, in order to express the k-point distribution along space-like paths in half-space LPP with geometric (Proposition 4.2) and exponential (Proposition 1.7) weights. In Section5, we discussed the issues related to the multiplicity of the limiting point process, and prove the first part of Theorem 1.4. In Section6, we perform all other asymptotic analysis: we prove limit theorems towards the GUE Tracy-Widom distribution (Theorem 1.5), GOE Tracy-Widom distribution (second part of Theorem 1.4), and Gaussian distribution (third part of 1.4).

Acknowledgements. The authors would like to thank Alexei Borodin and Eric Rains for sharing their insights about Pfaffian Schur processes. We are especially grateful to Alexei Borodin for discussions related to the dynamics preserving Pfaffian Schur processes and for drawing our attention to the reference [22]. We also thank Tomohiro Sasamoto and Takashi Imamura for discussions regarding their paper [38]. Part of this work was done during the stay of J. Baik, G. Barraquand and I. Corwin at the KITP and supported by NSF PHY11-25915. J. Baik was supported in part by NSF grant DMS-1361782. G. Barraquand was supported in part by the LPMA UMR CNRS 7599, UniversitéParis-Diderot–Paris 7 and the Packard Foundation through I. Corwin’s Packard Fellowship. I.Corwin was supported in part by the NSF through DMS-1208998, a Clay Research Fellowship, the PoincaréChair, and a Packard Fellowship for Science and Engineering.

4 The integration domain in [25, (4.44)] should be R<0 instead of R>0. The correct formula was found in [38, (5.39)]. 7 2. Fredholm Pfaffian formulas Let us introduce a convenient notation that we use throughout the paper to specify integration contours in the complex plane. ϕ Deﬁnition 2.1. Let Ca be the union of two semi-inﬁnite rays departing a ∈ C with angles ϕ and −ϕ. We assume that the contour is oriented from a + ∞e−iϕ to a + ∞e+iϕ.

2.1. GUE Tracy-Widom distribution. For a kernel K : X × X → R, we deﬁne its Fredholm determinant det(I + K)L2(X,µ) as given by the series expansion ∞ Z Z X 1 k ⊗k det(I + K) 2 = 1 + ··· det K(x , x ) dµ (x . . . x ), L (X,µ) k! i j i,j=1 1 k k=1 X X whenever it converges. We will generally omit the measure µ in the notations and write simply 2 L (X) when the uniform or the Lebesgue measure is considered.

Deﬁnition 2.2. The GUE Tracy-Widom distribution, denoted LGUE is a probability distribution on R such that if X ∼ LGUE,

P(X 6 x) = FGUE(x) = det(I − KAi)L2(x,+∞) where KAi is the Airy kernel, Z Z ez3/3−zu 1 KAi(u, v) = dw dz 3 . (1) 2π/3 π/3 ew /3−wv z − w C−1 C1 Throughout the paper, all integrals over a contour of the complex plane will take a factor 1/(2iπ), 1 and thus we use the notation dz := 2iπ dz. 2.2. Fredholm Pfaffian. In order to define the GOE and GSE distribution in a form which is convenient for later purposes, we introduce the concept of Fredholm Pfaffian. We refer to Section 4.1 explaining how Fredholm Pfaffians naturally arise in the study of Pfaffian point processes. Definition 2.3 ([37] Section 8). For a 2 × 2-matrix valued skew-symmetric kernel, K11(x, y) K12(x, y) K(x, y) = , x, y ∈ X, K21(x, y) K22(x, y) we define its Fredholm Pfaffian of K by the series expansion ∞ Z Z k X 1 ⊗k Pf J + K 2 = 1 + ··· Pf K(xi, xj) dµ (x1 . . . xk), (2) L (X,µ) k! i,j=1 k=1 X X provided the series converges, and we recall that for an skew-symmetric 2k × 2k matrix A, its Pfaffian is defined by 1 X Pf(A) = sign(σ)a a . . . a . (3) 2kk! σ(1)σ(2) σ(3)σ(4) σ(2k−1)σ(2k) σ∈S2k The operator J is defined by its kernel J, where 0 1 J(x, y) = 1 . x=y −1 0

8 Remark 2.4. A 2 × 2-matrix valued kernel K(x, y) is called skew-symmetric if the 2k × 2k matrix k 2 K(xi, xj) is skew-symmetric. We must consider it as made up of k blocks, each of which i,j=1 has size 2 × 2. Considering 22 blocks of size k × k instead would change the value of its Pfaﬃan by a factor (−1)k(k−1)/2.

In Sections4,5 and6, we will need to control the convergence of Fredholm Pfaﬃan series expansions. This can be done using Hadamard’s bound. We recall that for a matrix M of size k, if the (i, j) entry of M is bounded by aibj for all i, j ∈ {1, . . . , k} then

k k/2 Y det[M] 6 k aibi. i=1

Using this inequality and the fact that Pf[A] = pdet[A] for a skew-symmetric matrix A, we have

Lemma 2.5. Let K(x, y) a 2 × 2 matrix valued skew symmetric kernel. Assume that there exist constants C > 0 and constants a > b > 0 such that −ax−ay −ax+by bx+by |K11(x, y)| < Ce , |K12(x, y)| < Ce , |K22(x, y)| < Ce .

Then, for all k ∈ Z>0,

k k Y PfK(x , x ) < (2k)k/2Ck e−(a−b)xi . i j i,j=1 i=1

2.3. GOE Tracy-Widom distribution. The GOE Tracy-Widom distribution, denoted LGOE, is a continuous probability distribution on R. The following is a convenient formula for its cumulative distribution function.

Lemma 2.6. For X ∼ LGOE, GOE FGOE(x) := (X x) = Pf J − K 2 , P 6 L (x,∞) where KGOE is the 2 × 2 matrix valued kernel deﬁned by Z Z GOE z − w z3/3+w3/3−xz−yw K11 (x, y) = dz dw e , π/3 π/3 z + w C1 C1 Z Z GOE GOE w − z z3/3+w3/3−xz−yw K12 (x, y) = −K21 (x, y) = dz dw e , π/3 π/3 2w(z + w) C1 C−1/2 Z Z GOE z − w z3/3+w3/3−xz−yw K22 (x, y) = dz dw e π/3 π/3 4zw(z + w) C1 C1 Z 3 dz Z 3 dz sgn(x − y) + ez /3−zx − ez /3−zy − . π/3 4z π/3 4z 4 C1 C1 Throughout the paper, we adopt the convention that

sgn(x − y) = 1x>y − 1x

9 Proof. According to [40] (see also [23, (2.9) and (6.17)]), the GOE distribution function can be 1 1 5 deﬁned by FGOE = Pf J − K , where K is deﬁned by the matrix kernel Z ∞ Z ∞ 1 0 0 K11(x, y) = dλAi(x + λ)Ai (y + λ) − dλAi(y + λ)Ai (x + λ), 0 0 Z ∞ Z ∞ 1 1 1 K12(x, y) = −K21(y, x) = Ai(x + λ)Ai(y + λ) + Ai(x) dλAi(y − λ) 0 2 0 Z ∞ Z ∞ Z ∞ Z ∞ 1 1 1 K22(x, y) = dλ dµAi(y + µ)Ai(x + λ) − dλ dµAi(x + µ)Ai(y + λ) 4 0 λ 4 0 λ 1 Z +∞ 1 Z +∞ sgn(x − y) − Ai(x + λ)dλ + Ai(y + λ)dλ − . 4 0 4 0 4 Using the contour integral representation of the Airy function Z z3/3−zx Ai(x) = e dz, for any ϕ ∈ (π/6, π/2] and a ∈ >0, (4) ϕ R Ca

π/3 0 with integration on C1 for Ai(x + λ) and Ai (x + λ), we readily see that ∞ Z Z Z 3 3 K1 (x, y) = dλ dz dw(z − w)ez /3+w /3−xz−yw−λz−λw 11 π/3 π/3 0 C1 C1 ∞ Z Z 3 3 Z = dz dw(z − w)ez /3+w /3−xz−yw dλe−λz−λw (5) π/3 π/3 C1 C1 0 GOE = K11 (x, y). The exchange of integrations in (5) is justified because z + w has positive real part along the GOE π/3 π/3 contours. Turning to K12 , we first deform the contour C−1/2 for the variable w to C1 . When doing this, we encounter a pole at zero. Thus, taking into account the residue at zero, we find that Z Z GOE 1 z − w z3/3+w3/3−xz−yw K12 (x, y) = Ai(x) + dz dw e 2 π/3 π/3 2w(z + w) C1 C1

π/3 Using again (4) with the contour C1 , we ﬁnd that Z ∞ Z ∞ GOE 1 1 1 K12 (x, y) = Ai(x) + Ai(x + λ)Ai(y + λ)dλ − Ai(x) dλAi(y + λ) = K12(x, y), 2 0 2 0

R +∞ 1 where in the last equality we have used that 0 Ai(λ)dλ = 1. For K22 we use similarly (4) with π/3 1 GOE integration on C1 for Ai(x + λ) and Ai(x + µ) and get that K22(x, y) = K22 (x, y).

2.4. GSE Tracy-Widom distribution. The GSE Tracy-Widom distribution, denoted LGSE, is a continuous probability distribution on R. Lemma 2.7. For X ∼ LGSE,

GSE FGSE (x) := (X x) = Pf J − K 2 , P 6 L (x,∞)

5Our formulas exactly correspond to [23]. The expression in [40] can be shown to be equivalent by multiplying the kernel on the right by J and using a conjugation which preserves the Pfaﬃan. 10 where KGSE is a 2 × 2-matrix valued kernel deﬁned by Z Z GSE z − w z3/3+w3/3−xz−yw K11 (x, y) = dz dw e , π/3 π/3 4zw(z + w) C1 C1 Z Z GSE GSE z − w z3/3+w3/3−xz−yw K12 (x, y) = −K21 (x, y) = dz dw e , π/3 π/3 4z(z + w) C1 C1 Z Z GSE z − w z3/3+w3/3−xz−yw K22 (x, y) = dz dw e . π/3 π/3 4(z + w) C1 C1

q 4 4 Proof. FGSE(x) is defined as FGSE(x) = det(I − K )L2(x,∞) in [40, Section III], where K is the 2 × 2 matrix valued kernel defined by Z ∞ 4 4 1 1 K11(x, y) = K22(y, x) = KAi(x, y) − Ai(x) Ai(λ)dλ, 2 4 y −1 1 K4 (x, y) = ∂ K (x, y) − Ai(x)Ai(y), 12 2 y Ai 4 Z ∞ Z ∞ Z ∞ 4 −1 1 K21(x, y) = KAi(λ, y)dλ + Ai(λ)dλ Ai(µ)dµ. 2 x 4 x y In order to have a Pfaffian formula, we need a skew-symmetric kernel. We compute the kernel KGSE := JK4 using 0 1 a b c d = , −1 0 c d −a −b and find that the operator KGSE is given by the matrix kernel GSE 4 K11 (x, y) = K21(x, y), GSE 4 K12 (x, y) = K22(x, y), GSE 4 4 GSE K21 (x, y) = −K11(x, y) = −K22(y, x) = −K12 (y, x), GSE 4 K22 (x, y) = −K12(x, y). π/3 Using the contour integral representations for the Airy function (4) with integrations on C1 and GSE the definition of the Airy kernel (1), we find that the entries Kij (x, y) match those given in the statement of Lemma 2.7. Now KGSE is skew-symmetric. Using that for a skew-symmetric kernel A, Pf(J + A)2 = det(I − JA) as soon as Fredholm expansions are convergent ([37, Lemma 8.1] , see also [35, Proposition B.4] for a proof), we have that (using J 2 = I) q 4 GSE FGSE(x) = det(I − K ) 2 = Pf J − K 2 . L (x,∞) L (x,∞) Remark 2.8. There exists an alternative scalar kernel which yields the GSE distribution function [28]: GSE q GLD FGSE(x) = Pf J − K ) 2 = det I − K 2 , (6) L (x,∞) L (x,∞) where Z ∞ GLD 1 K (x, y) = KAi(x, y) − Ai(x) dz Ai(y + z). 2 0 The identity (6) can be shown by factoring KGSE and using the det(I + AB) = det(I + BA) trick. 11 2.5. Crossover kernels. The crossover kernel in Theorem 1.8 concerns the limiting fluctuations of multiple points, hence the kernel is indexed by elements of {1, . . . , k}×R. We introduce a matrix cross kernel K , depending on parameters $ ∈ R and 0 6 η1 < ··· < ηk, which decomposes as Kcross(i, x; j, y) = Icross(i, x; j, y) + Rcross(i, x; j, y). We have Z Z cross z + ηi − w − ηj z + $ + ηi w + $ + ηj z3/3+w3/3−xz−yw I11 (i, x; j, y) = dz dw e , π/3 π/3 z + w + η + η z + η w + η C1 C1 i j i j Z Z cross z + ηi − w + ηj z + $ + ηi z3/3+w3/3−xz−yw I12 (i, x; j, y) = dz dw e , π/3 π/3 2(z + η )(z + η + w − η ) −w + $ + η Caz Caw i i j j cross cross I21 (i, x; j, y) = −I12 (j, y; i, x), Z Z z3/3+w3/3−xz−yw cross z − ηi − w + ηj e I22 (i, x; j, y) = dz dw . Cπ/3 Cπ/3 4(z − ηi + w − ηj) (z − $ − ηi)(w − $ − ηj) bz bw cross The contours in I12 are chosen so az > −ηi, az + aw > ηj − ηi and aw < $ + ηj. The contours in cross cross I22 are chosen so bz > ηi, bz > ηi + $ and bw > ηj, bw > ηj + $. We have R11 (i, x; j, y) = 0, cross and R12 (i, x; j, y) = 0 when i > j. When i < j, 4 2 2 − exp −(ηi−ηj ) +6(x+y)(ηi−ηj ) +3(x−y) cross 12(ηi−ηj ) R12 (i, x; j, y) = p , 4π(ηj − ηi) that we may also write Z +∞ cross −λ(ηi−ηj ) R12 (i, x; j, y) = − dλe Ai(xi + λ)Ai(xj + λ). −∞ cross The kernel R22 is antisymmetric, and when xi − ηi > xj − ηj we have

3 3 Z (z+ηi) /3+($+ηj ) /3−x(z+ηi)−y($+ηj ) cross −1 e R22 (i, x; j, y) = dz 4 π/3 $ + z Caz 3 3 1 Z e(z+ηj ) /3+($+ηi) /3−y(z+ηj )−x($+ηi) + dz 4 π/3 $ + z Caz 3 3 1 Z ze(z+ηi) /3+(−z+ηj ) /3−x(z+ηi)−y(−z+ηj ) − dz , 2 Cπ/3 ($ + z)($ − z) bz where the contours are chosen so that az < −$ and −|$| < bz < |$|.

Deﬁnition 2.9. We deﬁne the probability distribution Lcross by the distribution function cross Fcross(x; $, η) = Pf J − K (1, ·; 1, ·) . L2(x,∞)

This family of distribution functions realizes a two-dimensional crossover between GSE, GUE and GOE distributions, in the sense that Fcross(x; 0, 0) = FGOE(x) and  FGSE(x) when $ → +∞ and η = 0,  2  FGOE(x) when $ → −∞, η → +∞ with $/η = −1, Fcross(x; $, η) −→ FGUE(x) when η → +∞ and $ + η → +∞,  0 when $ → −∞ and $ + η → −∞. 12 Moreover, when $ → −∞ and $ + η → −∞, Z y 2 p 1 − 1 s2 Fcross(($ + η) + 2|$ + η|y; $, η) −→ √ e 2 ds. −∞ 2π In addition, for a ∈ R,

Fcross(x; $, η) −→ F1(x; a) when $ → −∞, η → +∞ with $ + η = a. 2 where F1(x; a) is the distribution in [3, Definition 1.3] that interpolates FGUE(x) and FGOE(x) . When η = 0, Fcross(x; $, 0) = F (x; $) where the crossover distribution F (x; w) is defined in [6, Definition 4]. The crossover kernel arising in Theorem 1.9 is a matrix kernel KSU, depending on parameters $ ∈ R and 0 < η1 < ··· < ηk, of the form KSU(i, x; j, y) = ISU(i, x; j, y) + RSU(i, x; j, y). We have Z Z z3/3+w3/3−xz−yw SU (z + ηi − w − ηj)e I11 (i, x; j, y) = dz dw , π/3 π/3 4(z + η )(w + η )(z + w + η + η ) C1 C1 i j i j Z Z z3/3+w3/3−xz−yw SU (z + ηi − w + ηj)e I12 (i, x; j, y) = dz dw , π/3 π/3 2(z + η )(z + w + η − η ) Caz Caw i i j SU SU I21 (i, x; j, y) = −I12 (j, xj; i, yi) Z Z SU z − ηi − w + ηj z3/3+w3/3−xz−yw I22 (i, x; j, y) = dz dw e , Cπ/3 Cπ/3 z − ηi + w − ηj bz bw SU SU The contours in I12 are chosen so az > −ηi, az + aw > ηj − ηi. The contours in I22 are chosen so that bz > ηi and bw > ηj. SU SU We have R11 (i, x; j, y) = 0, and R12 (i, x; j, y) = 0 when i > j. When i < j, 4 2 2 −(ηi−ηj ) +6(x+y)(ηi−ηj ) +3(x−y) 12(ηi−ηj ) SU −e cross R12 (i, x; j, y) = p = R12 (1, x; j, y). 4π(ηj − ηi) SU The kernel R22 is antisymmetric, and when x − ηi > y − ηj we have

1 Z 3 3 SU (z+ηi) /3+(−z+ηj ) /3−x(z+ηi)−y(−z+ηj ) R22 (i, x; j, y) = − dz ze , 2 π/3 C0 where the contours are chosen so that az > −$ and bz is between −$ and $. Modulo a conjugation of the kernel, KSU is the limit of Kcross when $ goes to +∞, i.e. 1 cross cross SU 4$2 K11 K12 K = lim cross 2 cross . $→∞ K21 4$ K22 3. Pfaffian Schur process The key tool in our analysis of last passage percolation on a half-space is the Pfaffian Schur process. In this Section, we first introduce the Pfaffian Schur process as in [21]. This is a Pfaffian analogue of the determinantal Schur process introduced in [34]. We refer to [19, Section 6] and references therein for background on the Schur process. Then, in order to connect it to last passage percolation, we introduce an equivalent presentation of the Pfaffian Schur processes indexed by lattice paths in the half-quadrant, and we study dynamics preserving the Pfaffian Schur process. 3.1. Definition of the Pfaffian Schur process. 13 3.1.1. Partitions and Schur positive specializations. Pfaffian Schur processes are measures on sequences of integer partitions. A partition is a nonincreasing sequence of nonnegative integers

λ = (λ1 > λ2 > ··· > λk > 0), with finitely many non-zero components. Given two partitions λ, µ, we write µ ⊂ λ if λi > µi for all i > 1. We write µ ≺ λ and say that µ interlaces λ if for all i > 1, λi > µi > λi−1. For a given partition λ, its dual, denoted λ0, is the partition corresponding to the Young diagram which is symmetric to the Young diagram of λ with respect to the first diagonal. In other words, 0 λi = ]{j : λj > i}. We say that a partition λ is even if all its component are even integers. Thus, 0 for a partition λ, we say that its dual λ is even when we have λ1 = λ2, λ3 = λ4, etc. We denote by |λ| the number of boxes in the diagram corresponding to λ. The probabilities in the Pfaffian Schur process – and the usual Schur process as well – are expressed in terms of skew Schur symmetric functions sλ/µ indexed by pairs of partitions µ, λ. We use the convention that sλ/µ = 0 if µ 6⊂ λ. We refer to [19, Section 2] for a definition of (skew) Schur functions and background on symmetric functions. We record later in Section 3.1.2 the few properties of Schur functions that we will use. A specialization ρ of the algebra Sym of symmetric functions is an algebra morphism from Sym to C. For example, the evaluation of a symmetric function into one fixed variable α ∈ C defines such a morphism. We denote f(ρ) the application of the specialization ρ to f ∈ Sym. A specialization can be defined by its values on a basis of Sym and we generally choose the power sum symmetric P n functions pn(x1, x2,... ) = xi . For two specializations ρ1 and ρ2, their union, denoted (ρ1, ρ2), is defined by

pn(ρ1, ρ2) = pn(ρ1) + pn(ρ2).

More generally, the specialization (ρ1, . . . , ρn) is the union of specializations ρ1, . . . , ρn. When ρ1 and ρ2 are specializations corresponding to the evaluation into sets of variables, then (ρ1, ρ2) corresponds to the evaluation into the union of both sets of variables, hence the term union for this operation on specializations.

Schur nonnegative specializations of Sym are specializations taking values in R>0 when applied to skew Schur functions sλ/µ for any partitions λ and µ. Thoma’s theorem (see [30] and references therein) provides a classiﬁcation of such specializations. Let α = αi i 1, β = βi i 1 and γ P > > be non-negative numbers such that (αi + βi) < ∞. Any Schur nonnegative specialization ρ is determined by parameters (α, β, γ) through the formal series identity

X n Y 1 + βiz hn(ρ)z = exp(γz) =: H(z; ρ). 1 − αiz n>0 i>1 where the hn are complete homogeneous symmetric functions. The specialization (0, 0, γ) is called (pure-)Plancherel and will be denoted by Plancherel(γ). It can be obtained as a limit of specializations (α, 0, 0) where α is the sequence of length M,(γ/M,...,γ/M), when M is sent to inﬁnity. From now on, we assume that all specializations are always Schur non-negative.

3.1.2. Useful identities. We collect some identities from [31] that were used in [21] to deﬁne the Pfaﬃan Schur process. The sums below (as in (7)) are always taken over all partitions, unless otherwise restricted. Starting with one already useful for the determinantal Schur process, we have the skew-Cauchy identity

X 0 0 X 0 sν/λ(ρ)sν/µ(ρ ) = H(ρ; ρ ) sλ/τ (ρ )sµ/τ (ρ), (7) ν τ 14 λ(1) λ(2) λ(n) + − + ρ+ − ρ0 ρ1 ρ1 n−1 ρn ... ∅ µ(1) µ(n−1) ∅

Figure 3. Diagram corresponding to The Pfaﬃan Schur process + + + − − PSP[ρ0 ; ρ1 ; ... ; ρn−1|ρ1 ; ... ; ρn ]. The ascending and descending links repre- sent inclusions of partitions.

0 where, if ρ and ρ are the specializations into sets of variables {xi}, {yi}, Y 1 H(ρ; ρ0) = . 1 − x y i,j i j We also use the branching rule for Schur functions X 0 0 sλ/ν(ρ)sν/µ(ρ ) = sλ/µ(ρ, ρ ). (8) ν The following identity is crucial to go from the determinantal Schur processes to its Pfaﬃan analog. It is a variant of the skew-Littlewood identity, X ◦ X sν/λ(ρ) = H (ρ) sλ/κ(ρ), (9) ν0even κ0even where if ρ is the specialization into a set of variables {xi}, Y 1 H◦(ρ) = . 1 − x x i

15 We may encode the choice of specializations by the diagram shown in Figure3. The normalization constant Z◦(ρ) for the weights V(λ,¯ µ¯) is X Z◦(ρ) := V(λ,¯ µ¯), ∅⊂λ(1)⊃µ(1)⊂λ(2)⊃µ(2)...µ(n−1)⊂λ(n)⊃∅ and can be computed [21] to be ◦ ◦ − − Y + − Z (ρ) = H (ρ1 , . . . , ρn ) H(ρi ; ρj ). 06i

(k) + + + − − Lemma 3.3. The law of λ under the Pfaffian Schur process PSP[ρ0 ; ρ1 ; ... ; ρn−1|ρ1 ; ... ; ρn ] + + − − is the Pfaffian Schur measure PSM[ρ0 , . . . ρk−1|ρk , . . . , ρn ]. Proof. Summing over all partitions except λ(k) in (10) and using identities (7), (8) and (9) yields the result. The following property of the Pfaffian Schur measure will be useful to interpret the results in Section 6.1. Proposition 3.4 (Corollary 7.6 [5]). Let c be a positive real and ρ a Schur nonnegative specialization. If µ is distributed according to PSM[c|ρ] and λ is distributed according to PSM[0|c, ρ], then (d) we have the equality in law λ1 = µ1. Proof. By definition, we have that

X τλ(0)sλ(c, ρ) X τµ(c)sµ(ρ) (λ x) = , (µ x) = . (11) P 1 6 Ho(c, ρ) P 1 6 Ho(ρ)H(c; ρ) λ:λ16x µ:µ16x o o The normalization constants H (c, ρ) and H (ρ)H(c; ρ) are equal. Since τλ(0) = 1λ0 even, we have that 1 X L.H.S.(11) = s (c, ρ) Ho(ρ)H(c; ρ) λ λ:λ16x λ0 even 1 X X = s (ρ)s (c) (Branching rule) Ho(ρ)H(c; ρ) µ λ/µ λ:λ16x µ λ0 even 1 X = s (ρ)cµ1−µ2+µ3−µ4+... Ho(ρ)H(c; ρ) µ µ:µ16x 1 X = s (ρ)τ (c) Ho(ρ)H(c; ρ) µ µ µ:µ16x = R.H.S.(11). 0 Note that in the third equality, we use that if λ is even and µ ≺ λ, then we have for all i > 1, λ2i−1 = µ2i−1 = λ2i. Thus, the fact that c is a single variable specialization is necessary. The fourth equality comes from the fact that for a single variable specialization c,

X µ1−µ2+µ3−µ4+... τµ(c) = sλ/κ(c) = c . (12) κ0 even 16 λ(4,4)

λ(5,3) λ(4,3)

∅ ∅ ∅

Figure 4. A zig-zag path indexing the Pfaﬃan Schur process.

Indeed, there is only one partition κ which gives a nonzero contribution in the sum in (12).

Remark 3.5. The proof of Prop. 3.4 actually shows that (λ1, λ3, λ5,... ) has the same law as (µ1, µ3, µ5,... ), but we will use only the first coordinates in applications. 3.2. Pfaffian Schur processes indexed by zig-zag paths. In Definition 3.1, Pfaffian Schur + + − − processes were determined by two sequences of specializations ρ◦ , . . . , ρn−1 and ρ1 , . . . , ρn . We describe here an equivalent way of representing Pfaffian Schur processes, indexing them by certain 2 zig-zag paths in the first quadrant of Z (as depicted in Figure4) where each edge of the path is labelled by some Schur nonnegative specialization. This representation is convenient for defining dynamics on Pfaffian Schur processes. More precisely, we consider oriented paths starting on the horizontal axis at (+∞, 0), proceeding by unit steps horizontally to the left or vertically to the top, hitting√ the diagonal at some point (n, n) for some n ∈ Z>0 and then taking one final edge of length n 2 to the origin (0, 0) (see Figure 4). Let us denote Ω the set of all such paths. For γ ∈ Ω, let us denote E(γ) its set of edges and V (γ) its set of vertices. We denote E↑(γ) the set of vertical edges and E←(γ) the set of horizontal edges. Each edge e is labelled by a Schur nonnegative specialization ρe. We label by a specialization v ρ◦ the edge joining the point (n, n) to (0, 0). Each vertex v ∈ γ indexes a partition λ . For γ ∈ Ω, the Pfaffian Schur process indexed by γ and the specializations on E(γ) is a probability v measure on the sequence of partitions λ := λ v∈V (γ) where the weight of λ is given by Y V(λ) = τλ(n,n) (ρ◦) W(e), e∈E↑(γ)∪E←(γ) ← ↑ where for an edge e1 ∈ E (π) and for an edge e2 ∈ E (π), v W e1 = v u = sλu/λv (ρe ) and W e2 = = sλv/λu (ρe ). 1 u 2 We adopt the convention that for all vertices v on the horizontal axis, λv is the empty partition, which in particular implies that specializations are empty on the horizontal axis. We also recall that sλ/µ ≡ 0 if µ 6⊂ λ. Using the above formalism, the normalisation constant associated to the weights V(λ) is given by ◦ Y ◦ Y Z (γ) = H (ρe)H(ρ◦; ρe) H(ρe; ρe0 ), e∈E↑(γ) e∈E↑(γ), e0∈E←(γ) e>e0 17 where e > e0 means that the edge e occurs after the edge e0 in the oriented path γ. Remark 3.6. The above description is equivalent to the one given in Definition 3.1. As we have defined in Section 3.1, the Pfaffian Schur process is a measure on sequences (1) (1) (2) (2) (n−1) (n) ∅ ⊂ λ ⊃ µ ⊂ λ ⊃ µ . . . µ ⊂ λ ⊃ ∅. However, by using empty specializations, one can force that λ(i) = µ(i) or µ(i) = λ(i+1). Hence, we can consider sequences where the order of inclusions is any word on the alphabet {⊂, ⊃}. By matching the inclusions ⊃ with vertical edges in the zig-zag path formulation and the inclusions ⊂ with horizontal edges, we see that both formulations are equivalent (zig-zag paths are in bijection with finite words over a two-letter alphabet). See for instance Figure6 for a particular example. This zig-zag construction also applies to the usual Schur process [34]. In that case, the processes are indexed by paths from a point on the horizontal axis to a point on the vertical axis. 3.3. Dynamics on Pfaffian Schur processes. Here we give a sequential construction of Pfaffian Schur processes using the formalism of Section 3.2 by defining Markov chains preserving the Pfaffian Schur structure. More precisely, we will define Markov dynamics such that the pushforward of the Pfaffian Schur process indexed by a path γ ∈ Ω and a set of specializations on E(γ) is the Pfaffian Shur process indexed by a path γ0 and the same set of specializations, where γ0 is obtained from γ by adding one box to the shape it defines, or half a box when it grows along the diagonal. These dynamics are a special case of those developed in [8] with Alexei Borodin in the Macdonald case [15]. They are an adaptation to the Pfaffian setting of Markov dynamics preserving the (determinantal) Schur process introduced in [17, Section 2] and studied in a more general setting in [13] (see the review [19, Section 6.4]). Let us explain precisely how to obtain a path γ ∈ Ω as the outcome of a sequence of elementary moves (see Figure5). ( i) We start with the path with only one vertex (0, 0). (ii) We may first make the path grow along the diagonal and get (1, 0) → (1, 1) → (0, 0). (iii) We can always freely move to the right the starting point of the path. For instance, assume that we get the new path (2, 0) → (1, 0) → (1, 1) → (0, 0). (iv) Then, we may make the path grow by one box and get (2, 0) → (2, 1) → (1, 1) → (0, 0). (v) We will continue iteratively adding boxes to the path or growing along the diagonal, until we arrive at γ. In a parallel way, we construct a sequence of Pfaffian Schur processes for each path described (0,0) above. (i) We start with an empty Pfaffian Schur process λ = ∅.(ii) The application of Markov dynamics – that we shall define momentarily – corresponding to the growth of the path (1,0) (1,1) (0,0) along the diagonal will define a Pfaffian Schur process on ∅ = λ ⊂ λ ⊃ λ = ∅. (iii) By convention, the partitions indexed by vertices on the horizontal axis are empty under the Pfaffian Schur process. Hence, sliding the starting point of the path to the right has no effect in terms of Pfaffian Schur process. (iv) Then, the application of Markov dynamics corresponding to the growth of the path by one box at the corner (1, 0) will define a Pfaffian Schur process (2,0) (2,1) (1,1) (0,0) ∅ = λ ⊂ λ ⊃ λ ⊃ λ = ∅.(v) We continue iteratively by applying Markov operators that update one partition in the Pfaffian Schur process. We now have to define the above-mentioned Markov dynamics, and explain how the specializations are chosen. We have seen that there are two distinct cases corresponding to the growth of the path by one box or the growth by a half box along the diagonal. x We define a transition operator Uρ1,ρ2 corresponding to the growth of the path by one box, as x a probability distribution Uρ1,ρ2 (π|ν, µ, κ) on partitions π, given partitions ν, µ, κ with µ ⊂ ν, κ. In terms of growing paths, this operator corresponds to adding one box in a corner formed by partitions µ ⊂ ν, κ, and it will update the partition µ to a partition π containing ν and κ, in such a way that the corner formed by partition κ, π, ν is the marginal of a new Pfaffian Schur process. x Pictorially, the action of the transition operator Uρ1,ρ2 can be represented by the following diagram. 18 λ(2,2)

λ(1,1) λ(1,1) λ(1,1) λ(2,1) λ(2,1)

λ(0,0) = ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ ∅ (i) (ii) (iii) (iv) (v)

Figure 5. Illustration of the ﬁrst steps according to which the path grows.

ρ2 ρ2 ν ? x ν π Uρ1,ρ2 ρ1 ρ1 ρ1 ρ1 µ κ µ κ ρ2 ρ2

x Uρ1,ρ2 updates the Pfaffian Schur process formed by partitions κ ⊃ µ ⊂ ν on the left to the the Pfaffian Schur process formed by partitions κ ⊂ π ⊃ ν on the right. After the update, on can forget the information supported by the gray arrows on the right. The specializations in the new Pfaffian Schur process are carried from the previous step by the transition operator as shown above.

x Lemma 3.7. Assume that Uρ1,ρ2 satisﬁes

X sπ/κ(ρ1)sπ/ν(ρ2) s (ρ )s (ρ )U x (π|ν, µ, κ) = . (13) κ/µ 2 ν/µ 1 ρ1,ρ2 H(ρ ; ρ ) µ 2 1 x 0 Then Uρ1,ρ2 preserves the Pfaﬃan Schur process measure in the following sense. Let γ ∈ Ω contain the corner deﬁned by the vertices ρ ρ (i + 1, j) −→2 (i, j) −→1 (i, j + 1), 00 where ρ1 and ρ2 are the specializations indexing edges of the corner. Let γ ∈ Ω contain the same set of vertex as γ0 except that the vertex (i, j) is replaced by the vertex (i + 1, j + 1), and assume that the specializations are now chosen as ρ ρ (i + 1, j) −→1 (i + 1, j + 1) −→2 (i, j + 1).

x (i+1,j+1) (i,j+1) (i,j) (i+1,j) 0 Then Uρ1,ρ2 (λ |λ , λ , λ ) maps the Pfaﬃan Schur process indexed by γ and the set of specializations on E(γ0) to the Pfaﬃan Schur process indexed by γ00 and the same set of specializations.

x 0 Proof. The relation (13) implies that applying Uρ1,ρ2 to the Pfaﬃan Schur process indexed by γ and averaging over λ(i,j) gives a weight proportional to

sλ(i+1,j+1)/λ(i+1,j) (ρ1)sλ(i+1,j+1)/λ(i,j+1) (ρ2), 00 as in the Pfaﬃan Schur process indexed by γ . The normalization H(ρ2; ρ1) accounts for the fact ◦ 00 ◦ 0 that Z (γ ) = H(ρ2; ρ1)Z (γ ). We choose a particular solution to (13) given by

s (ρ2)s (ρ1) U x (π|ν, µ, κ) = U x (π|ν, κ) = π/ν π/κ . (14) ρ1,ρ2 ρ1,ρ2 P λ sλ/ν(ρ2)sλ/κ(ρ1) The Cauchy identity (7) ensures that this choice satisfies (13). There exists other solutions to (13) (see [20, 32]) where U x(π|ν, µ, κ) depends on µ. 19 ∠ We also define a diagonal transition operator Uρ◦,ρ1 corresponding to the growth of the path ∠ by a half box along the diagonal as a probability distribution Uρ◦,ρ1 (π|κ, µ) on partitions π given partitions µ ⊂ κ. It will update the partition µ ⊂ κ to a partition π ⊃ κ such that π is a marginal of a new Pfaffian Schur process. Pictorially, ? π ρ1 ρ1 µ κ U ∠ ρ◦ κ ρ1 ρ◦,ρ1 ρ1 ρ◦

Again, the specializations in the new Pfaﬃan Schur process are carried from the previous step by the transition operator.

∠ Lemma 3.8. Assume that Uρ◦,ρ1 satisﬁes

X sπ/κ(ρ1)τπ(ρ◦) s (ρ )τ (ρ )U ∠ (π|κ, µ) = . (15) κ/µ 1 µ ◦ ρ◦,ρ1 H◦(ρ )H(ρ ; ρ ) µ 1 1 ◦ Let γ0 ∈ Ω contain the vertices (n, n) and (n + 1, n), with the edge (n + 1, n) → (n, n) labelled by 00 a specialization ρ1 and the diagonal edge labelled by ρ◦ as usual. Let γ ∈ Ω contain the vertices (n + 1, n + 1) and (n + 1, n), with the edge (n + 1, n) → (n + 1, n + 1) labelled by the specialization ρ1 (n+1,n+1) (n+1,n) (n,n) and the diagonal edge labelled by ρ◦. Then U ∠(λ |λ , λ ) maps the Pfaﬃan Schur process indexed by γ0 and the set of specializations on E(γ0) to the Pfaﬃan Schur process indexed by γ00 and the same set of specializations.

∠ 0 Proof. (15) implies that applying Uρ◦,ρ1 to the Pfaﬃan Schur process indexed by γ and averaging over λ(n,n) yields a weight proportional to

sλ(n+1,n+1)/λ(n+1,n) (ρ1)τλ(n+1,n+1) (ρ◦) 00 ◦ as in the Pfaﬃan Schur process indexed by γ . The normalization H (ρ1)H(ρ1; ρ◦) accounts for ◦ 00 ◦ ◦ 0 the fact that Z (γ ) = H (ρ1)H(ρ1; ρ◦)Z (γ ). We choose a particular solution to (15) given by

1 τπ(ρ◦)s (ρ1) U ∠ (π|κ, µ) = U ∠ (π|κ) = π/κ . (16) ρ◦,ρ1 ρ◦,ρ1 o H (ρ1)H(ρ1; ρ◦) τκ(ρ◦, ρ1) ◦ o Let us check that (15) is satisﬁed. Denoting Z = H (ρ1)H(ρ1; ρ◦),

X 1 X τπ(ρ◦)sπ/κ(ρ1) s (ρ )τ (ρ )U ∠ (π|κ, µ) = s (ρ )τ (ρ ) κ/µ 1 µ ◦ ρ◦,ρ1 Z◦ κ/µ 1 µ ◦ τ (ρ , ρ ) µ µ κ ◦ 1 P P τπ(ρ◦)sπ/κ(ρ1) ν0 even µ sκ/µ(ρ1)sµ/ν(ρ◦) = ◦ Z τκ(ρ◦, ρ1) P τπ(ρ◦)sπ/κ(ρ1) ν0 even sκ/ν(ρ1, ρ◦) = ◦ Z τκ(ρ◦, ρ1) τπ(ρ◦)s (ρ1) = π/κ . Z◦

∠ It is not a priori obvious that Uρ◦,ρ1 (π|κ) does deﬁne a probability distribution, but this can be checked using identities (9) and (7). 20 There may exist other solutions to (15) where U ∠(π|µ, κ) depends on µ. We do not attempt to classify other possible choices.

3.4. The ﬁrst coordinate marginal: last passage percolation. In this Section, we relate the dynamics on Pfaﬃan Schur process constructed in Section 3.3 to half-space LPP.

Deﬁnition 3.9 (Half-space geometric weight LPP). Let (ai)i>1 be a sequence of positive real 6 numbers and gn,m)n>m>0 be a sequence of i.i.d. geometric random variables with parameter anam when n > m + 1 and with parameter can when n = m. We deﬁne the geometric last passage percolation time on the half-quadrant (see Figure1), denoted G(n, m), by the recurrence for n > m, ( n o max G(n − 1, m),G(n, m − 1) if n > m + 1, G(n, m) = gn,m + G(n, m − 1) if n = m, with the boundary condition G(n, 0) ≡ 0.

Consider integers 0 < i1 < i2 < ··· < in and j1 > j2 > ··· > jn such that ik > jk for all k. The path k 7→ (ik, jk) is a down right path on the half-quadrant, it is called a space-like path in the context of interacting particle systems [16, 18]. We can express the joint law of the last passage times G(ik, jk) using a Pfaﬃan Schur process.

Proposition 3.10. Let c and a1, a2,... be positive real numbers. We have the equality in law

(d) (1) (n) G(i1, j1),...,G(in, jn) = λ1 , . . . , λ1 , where the sequence of partitions λ(1), . . . , λ(n) is distributed according the Pfaﬃan Schur process + + + − − PSP(ρ◦ ; ρ1 ; ... ; ρn−1|ρ1 ; ... ; ρn ) with + ρ◦ = (c, aj1+1, . . . , ai1 ), + ρk = (aik+1, . . . , aik+1 ) for k = 1, . . . , n − 1, − ρk = (ajk+1+1, . . . , ajk ) for k = 1, . . . , n − 1, − ρn = (a1, . . . , ajn ). Equivalently, in terms of zig-zag paths,

(d) (i1,j1) (in,jn) G(i1, j1),...,G(in, jn) = λ1 , . . . , λ1 where the sequence of partition λ(i1,j1), . . . , λ(in,jn) is distributed according to the Pfaﬃan Schur process indexed by a path γ ∈ Ω going through the points (i1, j1),..., (in, jn); where vertical (resp. horizontal) edges with horizontal (resp. vertical) coordinates i−1 and i are labelled by specializations into the single variable ai, and the diagonal edge is labelled by the specialization into the variable c (See Figure6).

(1) (2) Remark 3.11. It is also true that µ1 has the same law as G(i1, j2), µ1 has the same law as G(i2, j3), etc. but we do not use this fact. Proof. We begin with two lemmas explaining the action of the transition operators U x and U ∠ on the ﬁrst coordinates of the partitions.

6 The geometric distribution with parameter q ∈ (0, 1), denoted Geom(q), is the probability distribution on Z>0 k such that if X ∼ Geom(q), P(X = k) = (1 − q)q . 21 λ(3,3) λ(5,3) a4 a5

a3 (1) (2) (3) λ λ λ a a6 5 a 2 (6,2) ∅ , a λ , a 3 6 c 4 a c a 1

a2 ∅ µ(1) µ(2) ∅

Figure 6. Equivalence of the two formulations in Proposition 3.10. Left: The com- (i1,j1) (i2,j2) (i3,j3) ponents λ1 , λ1 , λ1 of the Pfaﬃan Schur process considered in Proposi- tion 3.10, for (i1, j1) = (3, 3), (i2, j2) = (5, 3) and (i3, j3) = (6, 2). Right: The corresponding diagram in the setting of Deﬁnition 3.1. There is equality in law (3,3) (5,3) (6,2) (1) (2) (3) between λ1 , λ1 , λ1 on the left, and λ , λ , λ on the right.

Lemma 3.12 ([17]). Consider two specializations determined by single variables a, b > 0 and the x transition operator Ua,b(π|ν, κ) deﬁned in (14) that updates randomly the partition µ to become π given partitions ν, κ. The ﬁrst coordinate of the partition π is such that

π1 = max(ν1, κ1) + Geom(ab). Proof. This Lemma is a particular case of two-dimensional push-block dynamics deﬁned in [17, |λ|−|µ| Section 2.6, Ex. (3)]. For two partitions λ, µ, we have that sλ/µ(a) = 1µ≺λa . Hence we have that x |π|−|ν| |π|−|κ| U (π|ν, κ) = 1ν≺π1κ≺πa b .

Summing out π2, π3,... and using the fact that U x is a stochastic transition kernel, we ﬁnd that the distribution of π1 is given by

U x(π |ν , κ ) = (1 − ab) · 1 abπ1−max(ν1,κ1). 1 1 1 π1>max(ν1,κ1) Lemma 3.13. Consider two specializations determined by single variables a, c > 0 and the tran- ∠ sition operator Uc,a(π|κ) deﬁned in (16) that updates randomly the partition µ to become π, given the partition κ. The ﬁrst coordinate of the partition π is such that

π1 = κ1 + Geom(ac). π −π π −π Proof. Similar to the proof of Lemma 3.12, we have that τπ(c) = c 1 2 c 3 4 ... Hence, the ∠ 1 π1−κ1 distribution of π1 under U (π|κ) is proportional to π1>κ1 (ac) . Fix some path γ ∈ Ω. Let us sequentially grow the Pfaffian Schur process according to the procedure defined in Section 3.3. We need to specify the diagonal specialization and the specializations we start from on the horizontal axis. Assume that the diagonal specialization is the single variable specialization into c > 0, and the specialization on the edge (i, 0) → (i − 1, 0) is the single variable specialization ai. Given how the specializations are transported on the edges of the lattice when we apply U x and U ∠, this choice of initial specializations implies that for all edges of γ, vertical (resp. horizontal) edges with horizontal (resp. vertical) coordinates i − 1 and i are labelled by specializations into the single variable ai, and the diagonal edge is labelled by the specialization into the variable c (See Figure6). 22 It follows from Lemmas 3.12 and 3.13 that the first coordinates of the partitions λv in the sequence of Pfaffian Schur processes are such that for i > j (i,j) (i−1,j) (i,j−1) λ1 = max λ1 , λ1 + Geom(aiaj), and (i,i) (i,i−1) λ1 = λ1 + Geom(aic). Hence, for any collection of vertices v1, . . . , vk along γ ∈ Ω, we have (d) v1 v1 (λ1 , . . . , λ1 ) = G(v1),...,G(v1) , where G(v) is the last passage time to v as in Definition 3.9. n Remark 3.14. Consider a symmetric matrix S = (gij)i,j=1 of size n whose entries gi,j are such that gi,j ∼ Geom(aiaj) for i 6= j and gi,i ∼ Geom(cai). The image of S under RSK correspondence is a Young tableau7 whose shape λ is distributed according to the Pfaffian Schur measure

PSM(c|a1, . . . , an). This can be deduced from [5, (7.47)], see also Eq. (10.151) in [24]. As a consequence, this provides a very short proof that G(n, n) has the law of the λ1 marginal of PSM(c|a1, . . . , an). Moreover, this provides an interpretation of the other coordinates of the partition: λ1+···+λi is the maximum over i-tuples of non intersecting up-right paths between the points (1, 1),..., (1, i) and (n, n),..., (n, n− i+1) of the sum of weights along these i paths. Note that the RSK insertion procedure also defines dynamics on (ascending) sequences of interlacing partitions, which are different from the push-block dynamics that we consider. Remark 3.15. Taking a Poinsson limit of geometric LPP, such that the points on the diagonal also limits to a one dimensional Poisson process, yields the Poissonized random involution longest increasing subsequence problem considered in [5,6] (equivalently half-space continuous PNG with a source at the origin [38]). The limit of Proposition 3.10 shows that this corresponds to the PLancherel specialization of the Pfaffian Schur process.

4. Fredholm Pfaffian formulas for k-point distributions It is shown in [21, Theorem 3.3] that if λ,¯ µ¯ is distributed according to a Pfaffian Schur process (Definition 3.1), then ¯ n (1) o n (k) o L(λ) := 1, λi − i ∪ · · · ∪ k, λi − i i>1 i>1 is a Pfaffian point process. In particular, for any S > 1 and pairwise distinct points (is, us), 8 1 6 s 6 S of {1, . . . , k} × Z we have the formal series identity S P {(i1, u1),..., (iS, uS)} ⊂ L(λ) = Pf K(is, us; it, ut) s,t=1, (17) where K(i, u; j, v) is a 2 × 2 matrix-valued kernel K K K = 11 12 K21 K22 T with K11, K12 = − K21 , and K22 given by explicit formulas (see [21, Theorem 3.3]). In Section 4.1, we first review the formalism of Pfaffian point processes. In Sections 4.2 and 4.4, we will

7Insertion tableau and Recording tableau are the same because S is symmetric. 8In [21] this is written as a formal identity in variables of the symmetric functions, but it can be specialized (as in Section 4.2) to numeric identities in the cases we consider. 23 explain how [21, Theorem 3.3] applies to last passage percolation in a half-quadrant with geometric or exponential weights.

4.1. Pfaffian point processes. We consider presently only simple9 point processes and introduce the formalism of Pfaffian point processes. Let us first start with the case of a finite state space X. A random (simple) point process on X is a probability measure on subsets X of X. The correlation functions of the point process are defined by ρ(Y ) = P X ⊂ X such that Y ⊂ X , for Y ⊂ X.

k This definition implies that for a domain I1 × I2 × · · · × Ik ⊂ X , X h i ρ(x1, . . . , xk) = E #{k-tuples of distinct points x1 ∈ X ∩ I1, . . . , xk ∈ X ∩ Ik} . (x1,...,xk)∈I1×···×Ik distinct (18) Such a point process is called Pfaffian if there exists a 2 × 2 matrix valued |X| × |X| skew-symmetric matrix K with rows and columns parametrized by points of X, such that the correlation functions of the random point process are ρ(Y ) = Pf KY , for any Y ⊂ X, (19) k where KY := K(yi, yj) i,j=1 for Y = {y1, . . . , yk}, and Pf is the Pfaffian (see (3)). More generally, let (X, µ) be a measure space, and P (and E) be a probability measure (and expectation) on the set Conf(X) of countable and locally finite subsets (configurations) X ⊂ X. We k define the kth factorial moment measure on X by h i B1 × · · · × Bk → E #{k-tuples of distinct points x1 ∈ X ∩ B1, . . . , xk ∈ X ∩ Bk} ,

⊗k where B1,...,Bk ⊂ X are Borel subsets. Assuming it is absolutely continuous with respect to µ , the k-point correlation function ρ(x1, . . . , xk) is the Radon-Nykodym derivative of the k-moment ⊗k factorial measure with respect to µ . As in the discrete setting, P is a Pfaﬃan point process if there exists a kernel K : X × X → Skew2(R), where Skew2(R) is the set of skew-symmetric 2 × 2 matrices, such that for all Y ⊂ X,

ρ(Y ) = Pf(KY ). (20) For measurable functions f : X → C, the definition of correlation functions is equivalent to Z h X i ⊗k E f(x1) . . . f(xk) = ρ(x1, . . . , xk)f(x1) . . . f(xk) dµ , (21) k k X (x1,...,xk)∈X distinct and thus (20) implies that ([37, Theorem 8.2]), h Y i E (1 + f(x)) = Pf J + K , (22) 2( ,fµ) x∈X L X whenever both sides are absolutely convergent. The R.H.S of (22) is a Fredholm Pfaffian, defined in Definition 2.3. We refer to [37, Section 8] and [35, Appendix B] for properties of Fredholm Pfaffians. In particular, (22) implies that the gap probabilities are given by the Pfaffian formula P no point lie in Y = Pf(J − K)L2(Y,µ) for Y ⊂ X. (23)

9i.e. without multiplicities 24 4.2. Half-space geometric weight LPP. Let a1, a2, · · · ∈ (0, 1), and c > 0 such that for all i, cai < 1; and let 0 < n1 < n2 < ··· < nk and m1 > m2 > ··· > mk be sequences of integers such that ni > mi for all i. Consider the Pfaﬃan Schur process indexed by the unique path in Ω going through the points

(nk, 0), (nk, mk), (nk−1, mk), (nk−1, mk−1),..., (n1, m1), (m1, m1), (0, 0), where a horizontal edge (i − 1, j) → (i, j) (resp. a vertical edge (i, j − 1) → (i, j)) is labelled by the specialization into the single variable ai (resp. aj), and the diagonal edge is labelled by c. In other words, we are in the same setting as in Section 3.4. In this case, it follows from [21, Theorem 3.3] that the correlation kernel of L(λ¯), that we will denote by Kgeo, indexed by couples of points in {1, . . . , k} × Z, is given by the following formulas. It is convenient to introduce the notations

ni nj mi mj geo Y z − a` Y w − a` Y 1 Y 1 h11 (z, w) := , z w 1 − a`z 1 − a`w `=1 `=1 `=1 `=1 ni nj mi mj geo Y z − a` Y 1 Y 1 Y w − a` h12 (z, w) := , z 1 − a`w 1 − a`z w `=1 `=1 `=1 `=1 ni nj mi mj geo Y 1 Y 1 Y z − a` Y w − a` h22 (z, w) := . 1 − a`z 1 − a`w z w `=1 `=1 `=1 `=1 Then the kernel is given by ZZ (z − w)hgeo(z, w) z − c w − c dz dw Kgeo(i, u; j, v) = 11 , (24) 11 (z2 − 1)(w2 − 1)(zw − 1) z w zu wv where the contours are positively oriented circles around 0 of such that for all i, 1 < |z|, |w| < 1/ai; ZZ (z − w)hgeo(z, w) z − c dz dw Kgeo(i, u; j, v) = 12 = −Kgeo(j, v; i, u), (25) 12 (z2 − 1)w(zw − 1) z(1 − cw) zu wv 21 where the contours are positively-oriented circles around 0 of radius such that for all i, 1 < |z| < 1/ai, |w| < 1/c, 1/ai and if i > j then |zw| > 1, while if i < j then |zw| < 1; ZZ (z − w)hgeo(z, w) 1 1 dz dw Kgeo(i, u; j, v) = 22 , (26) 22 zw(zw − 1) 1 − cz 1 − cw zu wv where the contours are positively oriented circles around 0 such that |zw| < 1 and |z|, |w| < 1/c, and for all i, |z|, |w| < 1/ai. The conditions on the contours ensure that (17) is not only a formal series identity but a numeric equality: when c < 1, all sums in the proof of [21, Theorem 3.3] are actually absolutely convergent, and the formulas can then be extended to any c such that cai < 1 by analytic continuation. The correlation functions of the model are clearly analytic in c, and the Pfaffians of integrals on finite contours are analytic as well. Using Proposition 3.10,(23) implies the following. Remark 4.1. In the statement of Theorem 3.3 in [21], the factor (zw − 1) which appear in the denominator of the integrand in K22 is replaced by (1 − zw). This seems to be a a typo, as the proof of Theorem 3.3 in [21] suggests that the correct factor should be (zw − 1), and [27] recently confirmed the correct sign.

Proposition 4.2. For any g1, . . . , gk ∈ Z>0, k ! \ geo P G(ni, mi) 6 gi = Pf J − K 2˜ , (27) L Dk(g1,...,gk) i=k 25 where ˜ Dk(g1, . . . , gk) = {(i, x) ∈ {1, . . . , k} × Z : x > gi}, and J is the matrix kernel 0 1 J(i, u; j, v) = 1 . (28) (i,u)=(j,v) −1 0

By Deﬁnition 2.3, PfJ − Kgeo is deﬁned by the expansion

∞ n k ∞ ∞ geo X (−1) X X X geo n Pf J − K 2˜ = 1 + ··· Pf K (is, xs; it, xt) . L Dk(g1,...,gk) n! s,t=1 n=1 i1,...,in=1 x1=gi1 xn=gin (29)

Remark 4.3. In the special case k = 1 and n1 = m1 = n, the Pfaffian representation of the (n,n) correlation functions for {λi − i} was given in [37, Corollary 4.3] and subsequently used in [26] to study involutions with a fixed number of fixed points. 4.3. Deformation of contours. We will later need (in order to deduce the correlation kernel of √ the exponential model from the geometric case) to set all ai to q and let q go to 1. Before taking this limit, we need to deform some of the contours used in the formulas for Kgeo and compute the residues involved. It will be much more convenient10 for the following asymptotic analysis if all integration contours in the definition of Kgeo are circles such that |zw| > 1. We do not need to geo geo transform the expressions for K11 and for K12 when i > j. When c < 1, these residues correspond to terms that arise in the Pfaffian version of Eynard-Mehta’s theorem ([21, Theorem 1.9]), as in the proof of Theorem 3.3 in [21]. When c > 1, we get more residues which were not present in the proof of Theorem 3.3 in [21] (recall that we make an analytic continuation to obtain the correlation kernel when c > 1). These extra residues are the signature of the occurrence of a phase transition when c varies, as discovered in [6]. geo Case c < 1: For K22 we write geo geo geo K22 (i, u; j, v) = I22 (i, u; j, v) + R22 (i, u; j, v) (30) geo where I22 (i, u; j, v) is the same as (26) but the contours are now positively oriented circles around 0 such that 1 < |z|, |w| < min(1/c, 1/a1, 1/a2,... ); and Z 1 − z2 1 1 hgeo(z, 1/z) Rgeo(i, u; j, v) = 22 dz. (31) 22 z2 1 − cz 1 − c/z zu−v To see the equivalence between (26) and (30), deform the w contour so that the radius exceeds z−1. Of course, we have to substract the residue for w = 1/z, which is expressed as an integral in geo z, which equals −R22 (i, u; j, v). Finally, we can freely move the z contour in the two-fold integral (only after having moved the w contour) and get (30). −1 In the case where i < j we need to also decompose K12, because we pick a residue at w = z geo when moving the w contour. We rewrite K12 as geo geo geo K12 (i, u; j, v) = I12 (i, u; j, v) + R12 (i, u; j, v) (32) geo where I12 (i, u; j, v) is the same as (25) but the contours are now positively-oriented circles around 0 of radius such that 1 < |z|, |w| < min(1/c, 1/a1, 1/a2,... ) and Z hgeo(z, 1/z) Rgeo(i, u; j, v) = − 12 dz. (33) 12 zu−v+1

10If it was not the case, we would find issues related to the choice of limiting contours when performing the asymptotic analysis. 26 Case c > 1: In that case, we need to take into account the residues at the poles of 1/(1 − cz) geo geo and 1/(1 − cw) in K12 and K22 . When deforming the w contour in (26), we first encounter a pole at 1/c and then a pole at 1/z. Each of these residues can be written as an integral in the variable z, where we may again deform the contour picking a residue at z = 1/c when necessary. Then we deform the contour for z in the two-fold integral and pick a residue at z = 1/c, which is expressed as an integral in w. We find geo geo ˆgeo K22 (i, u; j, v) = I22 (i, u; j, v) + R22 (i, u; j, v) (34) geo where I22 is as in the c < 1 case and Z hgeo(z, 1/c) Z hgeo(w, 1/c) Rˆgeo(i, u; j, v) = −cv 22 dz + cu 22 dw 22 zu+1(z − c) wv+1(w − c) Z 1 − z2 hgeo(z, 1/z) dz + 22 + cu−v−1hgeo(1/c, c). (35) z2 (1 − cz)(1 − c/z) zu−v 22 Because of the particular sequence of contour deformations that we have chosen, the first integral in (35) is such that c is outside the contour, while c is inside the contour in the second and third integral in (35). We get a formula where the antisymmetry is apparent by deforming again the z ˆgeo contour and writing R22 as Z hgeo(z, 1/c) Z hgeo(w, 1/c) Rˆgeo(i, u; j, v) = −cv 22 dz + cu 22 dw 22 zu+1(z − c) wv+1(w − c) Z 1 − z2 hgeo(z, 1/z) dz + 22 − cu−v−1hgeo(1/c, c) + cv−u−1hgeo(c, 1/c). (36) z2 (1 − cz)(1 − c/z) zu−v 22 22 where the contours are circles with radius between c and the 1/a`’s in the first and second integral, and radius between 1/c and c in the third integral. geo For K12 , in the case where i < j, we similarly rewrite geo geo ˆgeo K12 (i, u; j, v) = I12 (i, u; j, v) + R12 (i, u; j, v) geo where I12 is as in the c < 1 case and Z hgeo(z, 1/z) Z 1 − cz dz Rˆgeo(i, u; j, v) = − 12 dz − cw hgeo(z, 1/c) , (37) 12 zu−v+1 (z2 − 1)z 12 zu and the contours are circles around 0 such that 1 < |z| < min(1/a1, 1/a2,... ). Case c = 1: When deforming contours as previously, the sequence of residues encountered is slightly different than in the case c > 1, and we get: geo geo ¯geo K22 (i, u; j, v) = I22 (i, u; j, v) + R22 (i, u; j, v) (38) geo where I22 is as in the c < 1 case and −1 Z hgeo(z, 1) 1 Z hgeo(w, 1) R¯geo(i, u; j, v) = 22 dz + 22 dw 22 2iπ zu+1(z − 1) 2iπ wv+1(w − 1) 1 Z 1 + z hgeo(z, 1/z) − 22 dz − hgeo(1, 1). (39) 2iπ z2(1 − z) zu−v 22 where all contours are circles such that 1 < |z|, |w| < min(1/a1, 1/a2,... ). In the case where i < j, geo geo ¯geo K12 (i, u; j, v) = I12 (i, u; j, v) + R12 (i, u; j, v) geo where I12 is as in the c < 1 case and 1 Z hgeo(z, 1/z) 1 Z 1 hgeo(z, 1) R¯geo(i, u; j, v) = − 12 dz + 12 dz, (40) 12 2iπ zu−v+1 2iπ z(1 + z) zu 27 and the contours are circles around 0 such that |z| < min(1/a1, 1/a2,... ). 4.4. Half-space exponential weight LPP. The following lemma is a direct consequence of the geometric to exponential weak convergence. √ √ Lemma 4.4. Set all parameters ai ≡ q and c = q 1 + (α − 1)(q − 1) . Then, for any sequences of integers n1, . . . , nk and m1, . . . , mk such that ni > mi, we have the weak convergence as q → 1, k k (1 − q)G(ni, mi) i=1 =⇒ H(ni, mi) i=1. (Recall H(n, m) from Definition 1.1 and G(n, m) from Definition 3.9.) We will use this lemma to deduce the correlation functions for the exponential model. Define exp 2 the kernel K :({1, . . . , k} × R) → Skew2(R) by  Rexp(i, x; j, y) when α > 1/2,  Kexp(i, x; j, y) = Iexp(i, x; j, y) + Rêxp(i, x; j, y) when α < 1/2, R¯exp(i, x; j, y) when α = 1/2. Recalling Definition 2.1 for integration contours, we define Iexp by:

Z Z −xz−yw ni nj exp (z − w)e (1 + 2z) (1 + 2w) I11 (i, x; j, y) := dz dw (2z + 2α − 1)(2w + 2α − 1); π/3 π/3 4zw(z + w) (1 − 2z)mi (1 − 2w)mj C1/4 C1/4 (41) Z Z −xz−yw ni mj exp (z − w)e (1 + 2z) (1 + 2w) 2α − 1 + 2z I12 (i, x; j, y) := dz dw , (42) π/3 π/3 2z(z + w) (1 − 2w)nj (1 − 2z)mi 2α − 1 − 2w Caz Caw π/3 π/3 where in the definition of the contours Caz and Caw , the real constants az, aw are chosen so that 0 < az < 1/2, az + aw > 0 and aw < (2α − 1)/2; Z Z (z − w)e−xz−yw (1 + 2z)mi (1 + 2w)mj 1 1 Iexp(i, x; j, y) := dz dw , 22 n nj Cπ/3 Cπ/3 z + w (1 − 2z) i (1 − 2w) 2α − 1 − 2z 2α − 1 − 2w bz bw (43) where in the definition of the contours Cπ/3 and Cπ/3, the real constants b , b are chosen so that bz bw z w 0 < bz, bw < (2α − 1)/2 when α > 1/2, while we impose only bz, bw > 0 when α 6 1/2. exp exp êxp ¯exp We set R11 (i, x; j, y) = 0, and R12 (i, x; j, y) = 0 when i > j, and likewise for R and R . The other entries depend on the value of α and the sign of x − y. Case α > 1/2: When x > y,

Z mi mj −|x−y|z exp (1 + 2z) (1 − 2z) 2ze dz R22 (i, x; j, y) = − , π/3 (1 − 2z)ni (1 + 2z)nj (2α − 1 − 2z)2α − 1 + 2z Caz and when x < y

Z mj mi −|x−y|z exp (1 + 2z) (1 − 2z) 2ze dz R22 (i, x; j, y) = , π/3 (1 − 2z)nj (1 + 2z)ni (2α − 1 − 2z)(2α − 1 + 2z) Caz exp where (1 − 2α)/2 < az < (2α − 1)/2. One immediately checks that R22 is antisymmetric as we expect. When i < j and x > y

Z ni mj exp 1 (1 + 2z) (1 − 2z) −|x−y|z R12 (i, x; j, y) = − e dz, 2iπ π/3 (1 + 2z)nj (1 − 2z)mi C1/4 exp exp while if x < y, R12 (i, x; j, y) = R12 (i, y; j, x). Note that R12 is not antisymmetric nor symmetric (except when k = 1, i.e. for the one point distribution). 28 Case α < 1/2: When x > y, we have

1−2α y Z mi mj −xz êxp −e 2 (1 + 2z) (2α) e R22 (i, x; j, y) = dz 2 (1 − 2z)ni (2 − 2α)nj 2α − 1 + 2z 1−2α x m m −yz e 2 Z (1 + 2z) j (2α) i e + dz 2 (1 − 2z)nj (2 − 2α)ni 2α − 1 + 2z Z (1 + 2z)mi (1 − 2z)mj 1 1 − 2ze−|x−y|zdz π/3 (1 − 2z)ni (1 + 2z)nj 2α − 1 − 2z 2α − 1 + 2z Caz (x−y) 1−2α m m (y−x) 1−2α m m e 2 (2α) i (2 − 2α) j e 2 (2α) j (2 − 2α) i − + , (44) 4 (2 − 2α)ni (2α)nj 4 (2 − 2α)nj (2α)ni where the contours in the two first integrals pass to the right of (1 − 2α)/2. When x < y, the sign êxp êxp of the third term is flipped so that R22 (i, x; j, y) = −R22 (j, y; i, x). We can write slightly simpler formulas by reincorporating residues in the first two integrals: thus, when x > y,

1−2α y Z mi mj −xz ˆexp −e 2 (1 + 2z) (2α) e R22 (i, x; j, y) = dz 2 π/3 (1 − 2z)ni (2 − 2α)nj 2α − 1 + 2z Caz 1−2α x m m −yz e 2 Z (1 + 2z) j (2α) i e + dz 2 π/3 (1 − 2z)nj (2 − 2α)ni 2α − 1 + 2z Caz Z (1 + 2z)mi (1 − 2z)mj 1 1 − 2ze−|x−y|zdz, (45) π/3 (1 − 2z)ni (1 + 2z)nj 2α − 1 − 2z 2α − 1 + 2z Caz 2α−1 1−2α where 2 < az < 2 . When i < j, if x > y

Z ni mj êxp (1 + 2z) (1 − 2z) −|x−y|z R12 (i, x; j, y) = − e dz, (46) π/3 (1 + 2z)nj (1 − 2z)mi C1/4 êxp êxp while if x < y, R12 (i, x; j, y) = R12 (i, y; j, x). êxp ˆgeo Remark 4.5. The formula (46) for R12 is not exactly the limit of the expression for R12 in (37). Indeed, the second term in (37), which corresponds to a residue when w = 1/c does not have its counterpart in (46). This is because we have assumed that aw < (2α − 1)/2 in the contours for exp geo I12 in (42), while we had w > 1/c in the contours for I12 in the case c > 1. Case α = 1/2: When x > y,

Z mi −xz Z mj −yz ¯exp (1 + 2z) e (1 + 2z) e R22 (i, x; j, y) = − dz + dz π/3 (1 − 2z)ni 4z π/3 (1 − 2z)nj 4z C1/4 C1/4 Z (1 + 2z)mi (1 − 2z)mj e−|x−y|zdz 1 + − , (47) π/3 (1 − 2z)ni (1 + 2z)nj 2z 4 C1/4 ¯exp ¯exp with a modiﬁcation of the last two terms when x < y so that R22 (i, x; j, y) = −R22 (j, y; i, x). When i < j, if x > y

Z ni mj ¯exp (1 + 2z) (1 − 2z) −|x−y|z R12 (i, x; j, y) = − e dz, π/3 (1 + 2z)nj (1 − 2z)mi C1/4 ¯exp ¯exp while if x < y, R12 (i, x; j, y) = R12 (i, y; j, x). 29 π/3 1 1 √ 1/ q

Figure 7. The deformed contour C< (thick black line) and the contour C (dashed line). On the right, zoom on the portion of the contour which contributes to the limit.

Proof of Proposition 1.7. By Lemma 4.4, the passage times {H(ni, mi)}i are the limits when q goes to 1 of the passage times {G(ni, mi)}i, and we know the probability distribution function of {G(ni, mi)}i from (27). Thus, the proof proceeds with two steps: (1) Take the asymptotics of the geo correlation kernel K involved in the probability distribution of {G(ni, mi)}i (Section 4.2), under √ √ the scalings ai ≡ q and c = q 1 + (α − 1)(q − 1) , x = (1 − q)u, y = (1 − q)v; (2) Deduce the asymptotic behavior of PfJ + Kgeo from the pointwise asymptotics of Kgeo. Using the weak convergence of Lemma 4.4, we get the distribution of {H(ni, mi)}i. Step (1): We use Laplace’s method to take asymptotics of Kgeo(i, u; j, v) under the scalings above. geo geo geo Let us provide a detailed proof for K11 . The arguments are quite similar for K12 and K22 . We have √ √ ZZ (z − w)(z − c)(w − c) (z − q)ni (w − q)nj dzdw Kgeo(i, u; j, v) = √ √ , (48) 11 2 2 m mj n +1 nj +1 u v C2 (z − 1)(w − 1)(zw − 1) (1 − z q) i (1 − w q) z i w z w

1+q−1/2 where the contour C can be chosen as a circle around 0 with radius 2 . Under the scalings considered, the dominant term in the integrand is 1 1 = = exp (1 − q)−1(−x log(z) − y log(w)). zuwv z(1−q)−1xw(1−q)−1y Using Cauchy’s theorem, we can deform the integration contours as long as we do not cross any pole. Hence, we can deform the contour C to be the contour C< depicted in Figure7. It is 1+q−1/2 constituted of two rays departing 2 with angles ±π/3 in a neighbourhood of 1 of size (for some small > 0), and the rest of the contour is given by an arc of a circle centered at 0. Its radius 1+q−1/2 R > 2 is chosen so that the circle intersects with the endpoints of the two rays departing 1+q−1/2 1+q−1/2 2 with angles ±π/3. Along the contour C<, Re[− log(z)] attains its maximum at 2 . Thus, the contribution of integration along the portion of contour which is outside the -box around −1 1 is negligible in the limit. More precisely, it has a contribution of order O(e−d(1−q) ) for some d > 0. Now we estimate the contribution of the integration in the neighbourhood of 1. Let us rescale the variables around 1 by setting z = 1 +z ˜(1 − q) and w = 1 +w ˜(1 − q). Cauchy’s theorem provides us some freedom in the choice of contours, and we can assume that after the change of variables, the integration contours are given by two rays of length (1 − q)−1 departing 1/4 in the directions 30 π/3 and −π/3. When q → 1, we have the point-wise limits √ z− q z − c zu −→ ezx˜ , √ −→ 1 + 2˜z, √ −→ −1 + 2˜z + 2α 1− q 1 − q √ 1−z q 1 − zc wv −→ ewy˜ , √ −→ 1 − 2˜z, √ −→ −1 − 2˜z + 2α 1− q 1 − q Using Taylor expansions to a high enough order, the integral in (48) converges to the integral of the pointwise limit of the integrand in (48), modulo a O() error, uniformly in q. The kind of estimates needed to control the error are given with more details in a similar situation in Section 5.2 (see in particular Equations (74) and (75)). Moreover, since the limiting integrand has exponential decay in direction ±π/3, one can replace the finite integration contours by infinite contours going to ∞eiπ/3 and ∞e−iπ/3 with an error going to 0 as q goes to 1. Since the size of the box around 1 (on which we have restricted the integration) can be taken as small as we want, we finally get that

Kgeo(i, u : j, v) Z Z z˜ − w˜ √11 −−−→ dz˜ dw˜ e−xz˜−yw˜ (1 − q)ni+nj −mi−mj +2 q→1 π/3 π/3 4˜zw˜(˜z +w ˜) C1/4 C1/4 (1 + 2˜z)ni (1 + 2˜w)nj × (2˜z + 2α − 1)(2w ˜ + 2α − 1). (1 − 2˜z)mi (1 − 2w ˜)mj Likewise, we obtain

Igeo(i, u : j, v) Z Z z˜ − w˜ 12 √ −−−→ dz˜ dw˜ e−xz˜−yw˜ (1 − q)(1 − q)ni+mj −nj −mi q→1 π/3 π/3 2˜z(˜z +w ˜) Caz Caw (1 + 2˜z)ni (1 + 2w ˜)mj 2α − 1 + 2˜z × , (1 − 2w ˜)nj (1 − 2˜z)mi 2α − 1 − 2w ˜

π/3 π/3 where the new contours Caz and Caw are chosen as in (42). In the case where α < 1/2 or α = 1/2, π/3 geo the contour Caw is not exactly the limit of the contour in the definition of I12 in (32), but we can exclude the pole at w = (2α − 1)/2 (so that aw < (2α − 1)/2) and remove the corresponding êxp ¯exp residue in R12 and R12 (see Remark 4.5). Thus, we have that Rgeo(i, u; j, v) 12 √ −−−→ Rexp(i, x; j, y), (49) (1 − q)(1 − q)ni+mj −nj −mi q→1 12 ˆgeo ¯geo exp and likewise for R12 and R12 (modulo the residue that is reincorporated in I12 ). Convergence (49) needs some justification though: for x > y, we use the exact same arguments as above, since the factor 1/(zu−v) ≈ e−(x−y)˜z in (33) has exponential decay along the tails of the contour. For x < y, we make the change of variables z = 1/zˆ and we adapt the case x > y. This shows that Kgeo(i, u; j, v) 12 √ −−−→ Kexp(i, x; jy). (1 − q)(1 − q)ni+mj −nj −mi q→1 12 Using very similar arguments, we find that

Igeo(i, u; j, v) Z Z z˜ − w˜ 22 √ −−−→ dz˜ dw˜ e−xz˜−yw˜ 2 mi+mj −ni−nj −2 (1 − q) (1 − q) q→1 Cπ/3 Cπ/3 z˜ +w ˜ bz bz (1 + 2˜z)mi (1 + 2w ˜)mj 1 1 × , (1 − 2˜z)mi (1 − 2w ˜)mj 2α − 1 − 2˜z 2α − 1 − 2w ˜ 31 where Cπ/3 and Cπ/3 are chosen as in (43), and bz bw Rgeo(i, u; j, v) 22√ −−−→ Rexp(i, x; jy), (1 − q)2(1 − q)mi+mj −ni−nj −2 q→1 22 and similar limits hold for Rˆgeo and R¯geo. 22 22 √ √ Note that when taking the Pfaffian, the factors (1− q)ni+nj −mi−mj +2 and (1− q)mi+mj −ni−nj −2 cancel out each other. This is because one can multiply the columns with odd index by (1 − √ √ q)mi+mj −ni−nj −2 and then the rows with even index by (1 − q)ni+nj −mi−mj +2, without chang- 11 ing the value of the Pfaffian . In the end, we find that for all integer n > 1 , positive real variables x1, . . . , xn, and integers 1 6 i1, . . . , in 6 k, n h geo −1 −1 i Pf K is, (1 − q) xs; it, (1 − q) xt n s,t=1 h exp i −−−→ Pf K is, xs; it, xt (1 − q)n q→1 s,t=1 Step (2): We need to prove that

k +∞ +∞ n X X X h geo i ··· Pf K is, us; it, ut s,t=1 i ,...,i =1 −1 −1 1 n u1=bgi1 (1−q) c un=bgin (1−q) c k Z +∞ Z +∞ n X h exp i −−−→ ··· Pf K is, xs; it, xt dx1 ... dxk. (50) q→1 s,t=1 gi gi i1,...,in=1 1 n For that purpose we need to control the asymptotic behaviour of the kernel Kgeo. 1 1−2α Lemma 4.6. There exists positive constants C > 0 and 2 > c1 > c2 > 2 such that uniformly for q in a neighbourhood of 1, for all x, y > 0, geo −1 −1 K11 (i, (1 − q) x; j, (1 − q) y) −c (x+y) √ 6 Ce 1 , (51) (1 − q)ni+nj −mi−mj geo −1 −1 K12 (i, (1 − q) x; j, (1 − q) y) −xc +yc √ 6 Ce 1 2 , (52) (1 − q)ni+nj −mi−mj geo −1 −1 K22 (i, (1 − q) x; j, (1 − q) y) xc +yc √ 6 Ce 2 2 . (53) (1 − q)ni+nj −mi−mj

Proof. The arguments are similar for all integrals involved. It amounts to justify that we can restrict the integration on a region on the complex plane with real part greater than c1 or −c2. Let us ﬁrst see how it works for the bound (51). We start with (48) as in Step (1). The contribution of integration along the circular parts of C< can be bounded by a constant for q in a neighbourhood of 1. Next we perform the change of variables z = 1 +z ˜(1 − q) and w = 1 +w ˜(1 − q) as in Step (1) and observe that all factors in the integrand are bounded for q in a neighbourhood of 1, except the factor 1/(zuwv) which behaves as 1 = exp − x(1 − q)−1 log(1 +z ˜(1 − q)) − y(1 − q)−1 log(1 +w ˜(1 − q)) ≈ e−xz˜−yw˜. zuwv

Since we can deform the contours so that Re[˜z] and Re[w ˜] are greater than c1 (for some c1 < 1/2), we get the desired bound for K11. Strictly speaking, Cauchy’s theorem allows to deform the contour

11Otherwise said, we conjugate the kernel by a diagonal kernel with determinant 1. 32 C< so that the real part stays above c1 along the two rays, and then on cuts the circular part which has a negligible contribution. Similarly, we find that geo −1 −1 I12 (i, (1 − q) x; j, (1 − q) y) −xc +yc √ 6 Ce 1 2 (1 − q)ni+nj −mi−mj by restricting the integration on contours such that Re[˜z] > c1 and Re[w ˜] > −c2. When deforming 2α−1 1−2α 1 the contour for w, we have to take into account the pole at 2 . Since for any α > 0, 2 < 2 , 1 1−2α we can always find suitable constants c1, c2 > 0 such that 2 > c1 > c2 > 2 . By the same kind of arguments, ˆgeo −1 −1 R12 (i, (1 − q) x; j, (1 − q) y) −c|x−y| −xc +yc √ 6 Ce < Ce 1 2 , (1 − q)ni+nj −mi−mj ¯geo geo and it is easier to check that the inequality also holds for R12 and R12 , so that (52) holds. For K22, one does not need exponential decay. We have that geo −1 −1 I22 (i, (1 − q) x; j, (1 − q) y) √ 6 C (1 − q)ni+nj −mi−mj and ˆgeo −1 −1 R22 (i, (1 − q) x; j, (1 − q) y) |x−y| 1−2α xc +yc √ 6 C max 1; e 2 < Ce 2 2 , (1 − q)ni+nj −mi−mj so that (53) holds. Hence using Lemma 2.5, h in geo −1 −1 n/2 n −(c1−c2)(x1+···+xn) Pf K is, (1 − q) xs; it, (1 − q) xt 6 (2n) C e . s,t=1 and (50) follows by dominated convergence. Dominated convergence also proves the convergence of Fredholm Pfaffian series expansions, so that geo exp Pf J − K 2˜ −1 −1 −−−→ Pf J − K 2 . (54) ` Dk(h1(1−q) ,...,hk(1−q) ) q→1 L Dk(h1,...,hk) Finally, Proposition 4.4 implies that k ! k ! \ xi \ P G(ni, mi) 6 −−−→ P H(ni, mi) 6 xi . 1 − q q→1 i=1 i=1 which, together with (54), concludes the proof.

5. Convergence to the GSE Tracy-Widom distribution This section is devoted to the proof of the first part of Theorem 1.4, and thus we assume that α > 1/2. By Proposition 1.7, the probability P (H(n, n) < x) can be expressed as the Fredholm Pfaffian of the kernel Kexp. The most natural would be to prove that under the scalings considered, Kexp converges pointwise to KGSE, and that the convergence of the Fredholm Pfaffian holds as well, using estimates to control the absolute convergence of the series expansion. Unfortunately, this approach cannot work here because Kexp does not converge to KGSE. The next remark provides a heuristic explanation. Remark 5.1. We expect the kernel Kexp(1, x; 1, y) to be the Pfaffian correlation kernel of some (n) (n) (n) GSE simple Pfaffian point process, say X := X1 > ··· > Xn . On the other hand, K (x, y) is the correlation kernel of a Pfaffian point process χ1 > χ2 > . . . having the same law as the first eigenvalues of a matrix from the GSE in the large size limit. 33 (n) (n) (n) In light of Remark 3.14, the points Xi are such that X1 + ··· + Xi has the same law as the maximal weight of a i-tuple of non-intersecting up-right paths in a symmetric exponential (n) environment with weights E(α) on the diagonal and E(1) off-diagonal (in particular X1 has the same distribution as H(n, n) since the half-space LPP and symmetrized LPP are equivalent). When (n) α is large enough, it is clear that the optimal paths will seldom visit the diagonal, so that X1 and (n) 1/3 (n) X2 will have (under n scaling) the same scaling limit. More precisely, we expect that X2i−1 (n) (n) and X2i will both converge to χi, so that the point process X converges as n goes to infinity to a non-simple point process where each point has multiplicity 2. GSE (n) Let us denote by ρ the correlation function of the point process χ1, χ2,... , and let ρ be the (n) correlation function of the rescaled point process X −4n . By the characterization of correlation 24/3n1/3 functions in (21), we expect that for distinct points x1, . . . , xk, (n) k GSE ρ (x1, . . . , xk) −−−→ 2 ρ (x1, . . . , xk), n→∞ (n) and a singularity should appear in the limit of ρ (x1, . . . , xk) for k > 1 in the neighborhood of xi = xj. 5.1. A formal representation of the GSE distribution. We introduce a kernel which is a scaling limit of Kexp and whose Fredholm Pfaffian corresponds to the GSE distribution. However, it is not the Pfaffian correlation kernel of the GSE point process. This kernel is distribution valued, and we show in Proposition 5.2 that its Fredholm Pfaffian is the same as that of KGSE, by expanding and reordering terms in the Fredholm Pfaffian. In fact, to avoid dealing with distribution valued kernels, we later prove an approximate prelimiting version of this result as Proposition 5.7. Let us first recall that by Lemma 2.7, the kernel KGSE has the form A(x, y) −∂ A(x, y) KGSE(x, y) = y −∂xA(x, y) ∂x∂yA(x, y) GSE where A(x, y) is the smooth and antisymmetric kernel K11 . We introduce the kernel ∞ A(x, y) −2∂yA(x, y) K (x, y) = 0 −2∂xA(x, y) 4∂x∂yA(x, y) + δ (x, y) 0 2 where δ is a distribution on R such that ZZ Z 0 f(x, y)δ (x, y)dxdy = ∂yf(x, y) − ∂xf(x, y) y=x dx, (55) for smooth and compactly supported test functions f. The quantity Z ∞ k Pf[K (xi, xj)]i,j=1dx1 ... dxk (56) Rk 0 makes perfect sense because all terms of the form δ (xi, xj) are integrated against smooth functions 0 of xi, xj with sufficient decay. Expanding the Pfaffian and using the definition of δ ,(56) collapses into a sum of j-fold integrals for j = k down to k − bk/2c, where the integrands are smooth functions. We can then form the Fredholm Pfaffian of K∞, and we will show that the Fredholm Pfaffian series expansion is absolutely convergent so that we can reorder terms, and ultimately recognize the Fredholm Pfaffian of another kernel. 2 Proposition 5.2. Let A : R → Skew2(R) be a kernel of the form A(x, y) −∂ A(x, y) A(x, y) = y , −∂xA(x, y) ∂x∂yA(x, y) 34 where A is smooth, antisymmetric, and A satisfies the decay hypotheses of Lemma 2.5. Let B be the kernel A(x, y) −2∂yA(x, y) B(x, y) = 0 . −2∂xA(x, y) 4∂x∂yA(x, y) + δ (x, y) Then for any x ∈ R, Pf[J − A]L2(x,+∞) = Pf[J − B]L2(x,+∞). (57) In particular, GSE ∞ Pf[J − K ]L2(x,+∞) = Pf[J − K ]L2(x,+∞). (58)

Proof of Proposition 5.2. In order to simplify the notations in this proof, we assume that all integrals are performed over the domain (x, +∞). Denoting Z Z m bm = dx1··· dxm Pf[B(xi, xj)]i,j=1, and Z Z k ak = dx1··· dxk Pf[A(xi, xj)]i,j=1, we have ∞ ∞ X (−1)k X (−1)m Pf[J − A] = a and Pf[J − B] = b . k! k m! m k=0 m=0 0 Expanding the Pfaffian and using the definition of δ , bm collapses into a sum of j-fold integrals for j (−1)m j = k down to k − bk/2c. Denote by bm the j-fold integral term in the expansion of m! bm. The key fact towards the proof of the proposition is that k X (−1)k bk = a . (59) k+i k! k i=0 This corresponds to the fact that k-fold integral terms match in the expansion of both sides of (57). Then we can write ∞ ∞ m ∞ k ∞ X (−1)m X X X X X (−1)k b = bk = bk = a , (60) m! m m k+i k! k m=0 m=0 k=bm/2c k=0 i=0 k=0 which is what we wanted to prove. In order to reorder terms in the third equality in (60) (using k Fubini’s theorem), we must check that the double series bk+i is absolutely convergent. Using Lemma 2.5, it is enough to check that ∞ k X X 1 k 2k−1(2k)k/2Ck, k! i k=0 i=0 is absolutely convergent, which is clearly the case (C is some constant depending on A(x, y)). The rest of this proof is devoted to the proof of (59). The matrix B can be seen as the sum of a smooth matrix with good decay properties and another matrix whose entries are generalized 0 functions of the type δ (xi, xj). The following minor summation formula will be useful: Lemma 5.3 ([39] Lemma 4.2). Let M,N be skew symmetric matrices of size n for some even n. Then, X |I|−#I/2 Pf[M + N] = (−1) Pf[MI ]Pf[NIc ], I⊂{1,...,n};#I even 35 where #I denotes the cardinality of I, |I| the sum of its elements, and Ic its complement in {1, . . . , n}. For I ⊂ {1, . . . , n}, MI denotes the matrix M restricted to rows and columns indexed by elements of I. We also use the following definition of the Pfaffian, valid for any antisymmetric matrix M of size 2n: X Pf[M] = (α)Mi1,j1 ··· Min,jn , (61)

α={(i1,j1),...,(in,jn)} where the sum runs over partitions into pairs of the set {1,..., 2n}, in the form α = (i1, j1),..., (in, jn) with i1 < ··· < in and il < jl, and the signature ε(α) of the partition α is given by the signature of the permutation 1 2 3 ... 2n . i1 j1 i2 . . . jn k Let us fix some k > 1 and compute bk+i for i = 1 to k. It is instructive to consider first the case i = 1. k k 0 Computation of bk+1: By definition bk+1 corresponds to the terms of degree 1 in δ in the Pfaffian expansion of

k+1 k+1 (−1) A11(xi, xj) 2A12(xi, xj) 0 0 Pf + 0 (k + 1)! 2A21(xi, xj) 4A22(xi, xj) 0 δ (xi, xj) i,j=1 after taking the integral over x1, . . . , xk+1. Using Lemma 5.3, we can write (−1)k+1 Z Z X h i bk = dx ··· dx δ0(x , x )(−1)2i+2j−1Pf A2bi,2bj , (62) k+1 (k + 1)! 1 k+1 i j k+1 16i

2bi,2bj where An denotes the 2(n − 1) × 2(n − 1) dimensional matrix which is the result of removing columns and rows 2i and 2j from A (x , x ) 2A (x , x )n 11 a b 12 a b . 2A21(xa, xb) 4A22(xa, xb) a,b=1 By using operations on rows and columns, we ﬁnd that all terms in the summation in the R.H.S. of (62) are equal modulo a permutation of the xi. Since we integrate the xi over a symmetric domain, these permutations are inconsequential so that (−1)k+1k(k + 1) Z Z bk = − dx ··· dx δ0(x , x )Pf A2ck,2(\k+1) . k+1 2(k + 1)! 1 k+1 k k+1 k+1 Lemma 5.4. The action of δ0 is such that Z Z 0 2ck,2(\k+1) k dx1··· dxk+1 δ (xk, xk+1) Pf Ak+1 = −2 ak.

2ck,2(\k+1) Proof. We will use the shorthand notation Mk+1 for the matrix Ak+1 , and Mk for the matrix 2ck,2(\k+1) formed from the matrix Ak+1 after swapping the column and row of indices 2k − 1 and 2k. The notation Mj for j = k, k + 1 means that the column and row where the variable xj appears 36 2ck,2(\k+1) is on the rightmost/bottommost position. We have Pf[Mk] = −Pf[Ak+1 ] and Pf[Mk+1] = 2ck2(\k+1) Pf[Ak+1 ]. Then, Z Z 0 2ck,2(\k+1) dx1··· dxk+1 δ (xk, xk+1)Pf Ak+1 Z Z = dx1··· dxk ∂x Pf [Mk] + ∂x Pf [Mk+1] . k xk+1=xk k+1 xk+1=xk Moreover, expanding the Pfaﬃan as in (29) or (61), we see that h i ∂x Pf [Mk] = Pf (∂x Mk) k k xk+1=xk xk+1=xk and h i ∂x Pf [Mk+1] = Pf ∂x Mk+1 . k+1 k+1 xk+1=xk xk+1=xk

Using the diﬀerential relations ∂xk A11(x1, xk) = −A12(x1, xk), ∂xk A11(xk, x1) = −A21(xk, x1) and ∂xk A21(x1, xk) = −A22(x1, xk), ∂xk A12(xk, x1) = −A22(xk, x1), we see that h i k−1 k Pf (∂x Mk) = −2 Pf[A(xi, xj)] , k xk+1=xk i,j=1 and we obtain similarly that h i k−1 k Pf ∂x Mk+1 = −2 Pf[A(xi, xj)] . k+1 xk+1=xk i,j=1 Thus, Z Z Z Z 0 2ck,2(\k+1) k k k dx1··· dxk+1 δ (xk, xk+1)Pf Ak+1 = −2 dx1··· dxk Pf[A(xi, xj)]i,j=1 = −2 ak.

Thus, Lemma 5.4 shows that (−1)k+1 bk = k2k−1a . k+1 k! k k k 0 Computation of bk+i in general: By deﬁnition bk+i corresponds to the terms of degree i in δ in the Pfaﬃan expansion of k+i k+i (−1) A11(xi, xj) 2A12(xi, xj) 0 0 Pf + 0 (k + i)! 2A21(xi, xj) 4A22(xi, xj) 0 δ (xi, xj) i,j=1 after taking the integral over x1, . . . , xk+i. Using Lemma 5.3, (−1)k+i Z Z X h i bk = dx ··· dx Pf[δ0(x , x )]2i (−1)2j1+···+2j2i−iPf A2cj1,...,2dj2i , k+i (k + i)! 1 k+i jr js r,s=1 k+i 16j1<···

Again, all terms give an equal contribution after integrating over the xi, so that (−1)k+i k + i Z Z bk = (−1)i dx ··· dx Pf[δ0(x , x )]k+i Pf A2(k\−i+1),...,2(\k+i) . k+i (k + i)! 2i 1 k+i r s r,s=k−i+1 k+i 37 0 k+i 0 The Pfaﬃan Pf[δ (xr, xs)]r,s=k−i+1 can be expanded into products of δ acting on disjoint pairs of variables. Each term in the expansion will give the same contribution after we have integrated over the xjs, in the sense of the following. Lemma 5.5. For any partition into pairs of the set {k − i + 1, . . . , k + i}, in the form α = (r1, s1),..., (ri, si) with r1 < ··· < ri and rj < sj, we have Z Z 0 0 2(k\−i+1),...,2(\k+i) dx1··· dxk+i δ (xr1 , xs1 ) . . . δ (xri , xsi )Pf Ak+i Z Z i Y 0 2(k\−i+1),...,2(\k+i) = ε(α) dx1··· dxk+i δ (xk−i+2j−1, xk−i+2j)Pf Ak+i . (64) j=1

Proof. In order to match both sides of (64), we make a change of variables so that the δ0 factors exactly match. This does not change the value of the integral. Then we have to exchange rows/columns in the Pfaﬃan. Each time that we exchange two adjacent rows/columns12, we factors out a (−1), hence the signature prefactor in the R.H.S of (64). Lemma 5.6. The action of δ0 is such that Z Z i Y 0 2(k\−i+1),...,2(\k+i) i k dx1··· dxk+i δ (xk−i+2j−1, xk−i+2j)Pf Ak+i = (−1) 2 ak. (65) j=1

Proof. We adapt the proof of Lemma 5.4. For j1 < ··· < ji such that

j1 ∈ {k − i + 1, k − i + 2}, j2 ∈ {k − i + 3, k − i + 4}, . . . , ji ∈ {k + i − 1, k + i} 2(k\−i+1),...,2(\k+i) we denote by Mj1,...,ji the matrix obtained from Ak+i where for all r, one has moved the column and row where xjr appears to position 2(k − i + r). With this deﬁnition, if jr is even then the column and row where xjr appears do not move, and if jr is odd then the column and row where xjr appears is swapped with the right/bottom neighbor. Hence we have that

k\−i+1,...,[k+i j1+···+ji Pf[Ak+i ] = (−1) Pf[Mj1,...,ji ]. By the deﬁnition of δ0, Z Z i Y 0 2(k\−i+1),...,2(\k+i) dx1··· dxk+i δ (xk−i+2j−1, xk−i+2j)Pf Ak+i j=1 Z Z i Y 2(k\−i+1),...,2(\k+i) = dx1··· dxk (∂xk−i+2j − ∂xk−i+2j−1 )Pf Ak+i . (66) x =···=x =x j=1 k+i k+1 k We can develop the product of partial derivatives so that

Z Z i X Y j` 2(\k−i+1,...,2(\k+i) (66) = dx1··· dxk (−1) ∂x Pf A j` k+i xk+i=···=xk+1=xk j1,...,ji `=1 Z Z X = dx1··· dxk ∂x . . . ∂x Pf [Mj ,...,j ] , j1 ji 1 i xk+i=···=xk+1=xk j1,...,ji

12Recall that in Pfaffian calculus, one always exchanges rows and columns simultaneously, to preserve the antisymmetry of the matrix. 38 where the sums runs over all sequences j1 < ··· < ji such that j1 ∈ {k − i + 1, k − i + 2}, j2 ∈ {k − i + 3, k − i + 4}, etc. For any such sequence j1, . . . , ji, the relations between coefficients in the matrix A implies that i k−i k ∂x . . . ∂x Pf [Mj ,...,j ] = (−1) 2 Pf[A(xr, xs)] . j1 ji 1 i r,s=1 xk+i=···=xk+1=xk Thus, we find that Z Z i k k i k (66) = (−1) dx1··· dxk 2 Pf[A(xr, xs)]r,s=1 = (−1) 2 ak.

(2i)! 0 k+i There are 2ii! terms in the expansion of the Pfaffian of the 2i × 2i matrix δ (xr, xs) r,s=k−i+1. Combining Lemmas 5.6 and 5.5, we have (−1)k+i k + i(2i)! (−1)k k bk = 2ka = 2k−i(−1)ia . k+i (k + i)! 2i 2ii! k k! i k Summing over i, k k X (−1)k X k (−1)k bk = 2k−i(−1)ia = a . k+i k! i k k! k i=0 i=0 Proposition 5.2 suggests that if the kernel Kexp converges to K∞ under some scalings, then the exp ∞ Fredholm Pfaffian of K should converge to FGSE. However, the kernel K is distribution valued, and we do not know an appropriate notion of convergence to distribution-valued kernels. The following provides an approximative version of Proposition 5.2 that provides a notion of convergence sufficient for the Fredholm Pfaffian to converge.

Proposition 5.7. Let (An(x, y))n∈ be a family of smooth and antisymmetric kernels such that Z>0 r s 2 for all r, s ∈ Z>0, ∂x∂yAn(x, y) has exponential decay at inﬁnity (in any direction of R ), uniformly for all n. Let An(x, y) −∂yAn(x, y) An(x, y) := −∂xAn(x, y) ∂x∂yAn(x, y) and An(x, y) −2∂yAn(x, y) Bn(x, y) := 0 −2∂xAn(x, y) 4∂x∂yAn(x, y) + δn(x, y) 2 be two families of kernels (R) → Skew2(R) such that (1) The family of kernels An(x, y) −2∂yAn(x, y) −2∂xAn(x, y) 4∂x∂yAn(x, y) satisfy, uniformly in n, the hypotheses of Lemma 2.5. (2) The kernel An(x, y) converges pointwise to A(x, y) −∂ A(x, y) A := y , −∂xA(x, y) ∂x∂yA(x, y) 2 a smooth kernel R → Skew2(R) (necessarily satisfying the hypotheses of Lemma 2.5). 2 r s (3) For any smooth function f : R → R such that ∂x∂yf(x, y) have exponential decay at inﬁnity for all r, s, we have ZZ Z 0 δn(x, y)f(x, y)dxdy − ∂yf(x, y) − ∂xf(x, y) dx =: cn(f) −−−→ 0. x=y n→∞ 39 Then for all x ∈ R, Pf[J − Bn] 2(x,∞) −−−→ Pf[J − A] 2(x,∞). L n→∞ L

Proof. We will follow the proof of Proposition 5.2, and make approximations when necessary. Let Z Z m bm(n) = dx1··· dxm Pf[Bn(xi, xj)]i,j=1, Z Z m am(n) = dx1··· dxm Pf[An(xi, xj)]i,j=1,

k (−1)m and denote by bm(n) the part of the expansion of m! bm(n) involving only terms of degree m − k 0 in δn (as in (63)). We need to justify the approximation k X (−1)k bk (n) ≈ a (n), k+i k! k i=0 and show that the error is summable and goes to zero. Let us replace A, B by An, Bn in the proof of Proposition 5.2. The only part of the proof which breaks down is when we apply the deﬁnition 0 0 0 of δ . However, using the hypothesis (3), we see that replacing δ by δn in Lemma 5.6 results in an error bounded by i i 2 (cn) |ak(n)| ˜k for i > 1 and the error is zero when i = 0. Let bk+i(n) be the quantity having the same expression k 0 0 ˜k as bk+i(n) where all occurrences of δn are replaced by δ . In other terms, bk+i(n) is given by (63) where the kernel is An instead of A. We have 1 k bk (n) − ˜bk (n) 2k−i2i(c )i|a (n)|. k+i k+i 6 k! i n k On one hand, ∞ ∞ m X (−1)m X X Pf[J − B ] = b (n) = bk (n), n m! m m m=0 m=0 k=bm/2c while on the other hand, ∞ ∞ k ∞ m X (−1)k X X X X Pf[J − A ] = a = ˜bk = ˜bk (n), n k! k k+i m k=0 k=0 i=0 m=0 k=bm/2c where we have used (59) from the proof of Proposition 5.2 in the second equality and Fubini’s Theorem in the third equality (which we can do by hypothesis (1) and Lemma 2.5). Thus, ∞ m X X 1 k Pf[J − B ] − Pf[J − A ] 2k(c )m−k|a (n)| n n 6 k! m − k n k m=0 k=bm/2c ∞ k ∞ X X 1 k X 2k = 2k(c )i|a (n)| = |a (n)| (1 + c )k − 1 , k! i n k k! k n k=0 i=1 k=0 Using the bound k k k (1 + cn) − 1 6 k log(1 + cn) (1 + cn) 6 kcn (1 + cn) , we ﬁnd that ∞ k X 2 k Pf[J − Bn] − Pf[J − An] |ak(n)|kcn (1 + cn) −−−→ 0 6 k! n→∞ k=0 40 Moreover, using the pointwise convergence of hypothesis (2) and the decay hypothesis (1) along with Lemma 2.5, dominated convergence shows that

Pf[J − An] −−−→ Pf[J − A]. n→∞

5.2. Asymptotic analysis of the kernel Kexp. Recall that we have assumed that α > 1/2 (as in the first part of Theorem 1.4). Let us first focus on pointwise convergence of Kexp(1, x; 1, y) as n = m goes to infinity. In the following, we write simply Kexp(x; y) instead of Kexp(1, x; 1, y). We expect that as n goes to infinity, H(n, n) is asymptotically equivalent to hn for some constant h, and fluctuates around hn in the n1/3 scale. Thus, it is natural to study the correlation kernel at a point r(x, y) := nh + n1/3σx, nh + n1/3σy, (67) where the constants h, σ will be fixed later. We introduce the rescaled kernel

 1 exp 1/3 exp  2 K11 r(x, y) σn K12 r(x, y) Kexp,n(x, y) := (2α−1) .  1/3 exp 2 2/3 2 exp  σn K21 r(x, y) σ n (2α − 1) K22 r(x, y)

By a change of variables in the Fredholm Pfaffian expansion of Kexp and a conjugation of the kernel preserving the Pfaffian, we get using Proposition 1.7 that H(n, n) − hn exp,n P y = Pf[J − K ] 2(y,∞). σn1/3 6 L Moreover, we write Kexp,n(x, y) = Iexp,n(x, y) + Rexp,n(x, y), exp,n according to the decomposition made in Section 4.4. The kernel I11 can be rewritten as Z Z exp,n z − w (2z + 2α − 1)(2w + 2α − 1) I11 (x, y) = dz dw π/3 π/3 4zw(z + w) (2α − 1)2 C1/4 C1/4 h i × exp n(f(z) + f(w)) − n1/3σ(xz + yw) , (68) where f(z) = −hz + log(1 + 2z) − log(1 − 2z). (69) Setting h = 4, we find that f 0(0) = f 00(0) = 0. Setting σ = 24/3, Taylor expansion of f around zero yields σ3 f(z) = z3 + O(z4). (70) 3 exp,n Similarly, the kernel I12 can be rewritten as Z Z exp,n 1/3 z − w 2α − 1 + 2z I12 (x, y) = σn dz dw π/3 π/3 2z(z + w) 2α − 1 − 2w C1/4 C0 h i × exp n(f(z) + f(w)) − n1/3σ(zx + wy) . (71)

41 Here the assumption that α > 1/2 matters in the choice of contours, so that the poles for w = exp,n (2α − 1)/2 lie on the right of the contour. The kernel I22 can be rewritten as Z Z exp,n 2 2/3 z − w (2α − 1) (2α − 1) I22 (x, y) = σ n dz dw π/3 π/3 z + w 2α − 1 − 2z 2α − 1 − 2w C1/4 C1/4 h i times exp n(f(z) + f(w)) − n1/3σ(zx + wy) .

exp,n exp,n exp,n exp,n The kernel R is such that R11 = R12 = R21 = 0 and 2 exp,n 1/3 2α − 1 −|x−y|σn1/3 2α−1 R (x, y) = sgn(y − x) σn e 2 . 22 2 Proposition 5.8. For α > 1/2 and σ = 24/3, we have Z Z z3/3+w3/3−xz−yw exp,n GSE (z − w)e I11 (x, y) −−−→K11 (x, y) = dz dw , n→∞ π/3 π/3 4zw(z + w) C1 C1 Z Z z3/3+w3/3−xz−yw exp,n GSE (z − w)e I12 (x, y) −−−→2K12 (x, y) = dz dw , n→∞ π/3 π/3 2z(z + w) C1 C1 Z Z z3/3+w3/3−xz−yw exp,n GSE (z − w)e I22 (x, y) −−−→4K22 (x, y) = dz dw . n→∞ π/3 π/3 z + w C1 C1

exp,n Proof. Let us provide a detailed proof for the asymptotics of I11 . The arguments are almost exp,n exp,n identical for I12 and I22 , the only diﬀerence being the presence of poles that are harmless for the contour deformation that we will perform (see the end of the proof). We use Laplace’s method on the contour integrals in (68). It is clear that the asymptotic behaviour of the integrand is dominated by the term exp n(f(z) + f(w)). Let us study the π/3 behaviour of Re[f(z)] along the contour C0 .

π/3 13 Lemma 5.9. The contour C0 is steep descent for Re[f] in the sense that the functions t 7→ Re[f(teiπ/3)] and t 7→ Re[f(te−iπ/3)] are strictly decreasing for t > 0. Moreover, for t > 1, Re[f(te±iπ/3)] < −2t + 2. Proof. We have 2t Re[f(teiπ/3)] = Re[f(te−iπ/3)] = −2t + tanh−1 , 1 + 4t2 so that d 16(t2 + 2t4) Re[f(te±iπ/3)] = − , dt 1 + 4t2 + 16t4) which is always negative, and less than −2 for t > 1. π/3 We would like to deform the integration contour in (68) to be the steep-descent contour C0 . π/3 −1/3 This is not possible due to the pole at 0, and hence we modify the contour C0 in a n – neighbourhood of 0 in order to avoid the pole. We call C06∈ the resulting contour, which is depicted in Figure8. For a ﬁxed but large enough n, the integrand in (68) has exponential decay along the

13In general, we say that a contour C is steep descent for a real valued function g if g attains a unique maximum on the contour and is monotone along both parts of the contours separated by this maximum. 42 π/3 π/3

n−1/3 n−1/3 0 0

local C06∈ C06∈

local Figure 8. Contours used in the proof of Proposition 5.8. Left: the contour C06∈ . Right: the contour C06∈. tails of the integration contour: this is because for large n, the behaviour is governed by enf(z)+nf(w) and we have that Re[f(z)] −→ −∞ for z → ∞eiπ/3. |z| Hence, we are allowed to deform the contour to be C06∈ and write Z Z exp,n z − w (2z + 2α − 1)(2w + 2α − 1) I11 (x, y) = dz dw 2 C06∈ C06∈ 4zw(z + w) (2α − 1) h i × exp n(f(z) + f(w)) − n1/3σ(xz + yw) . (72) We recall that x and y here can be negative. We will estimate (72) by cutting it into two parts: The contribution of the integrations in a neighbourhood of 0 will converge to the desired limit, and the integrations along the rest of the contour will go to 0 as n goes to infinity. local Let us denote C06∈ the intersection of the contour C06∈ with a ball of radius centered at 0, and tail local tail set C06∈ := C06∈ \C06∈ . We first show that the R.H.S. of (72) with integrations over C06∈ goes to zero as n → ∞ for any fixed > 0. Using Lemma 5.9, we know that for any fixed > 0, there exists a constant c > 0 such that for t > , Re[f(te±iπ/3)] < −ct. Moreover, using both Taylor approximation (70) and Lemma 5.9, we know that for any z ∈ C06∈, Re[nf(z)] is bounded uniformly in n on the circular part of C06∈. This implies that there exists constants n0,C0 > 0 such that for tail n > n0 and z, w ∈ C06∈ we have

h 1/3 i −cn|z| exp n(f(z) + f(w)) − n σ(xz + yw) < C0e .

tail Using this estimate inside the integrand in (72) and integrating over C06∈ yields a bound of the form Ce−cn/2 which goes to 0 as n → ∞. local Now we estimate the contribution of the integration over C06∈ . We make the change of variables z = n−1/3z,˜ w = n−1/3w˜ and approximate the integrand in (72) by Z Z 3 3 z˜ − w˜ σ z˜3+ σ w˜3−σxz˜−σyw˜ e 3 3 d˜zdw. ˜ (73) 1/3 local 1/3 local 4˜zw˜(˜z +w ˜) n C06∈ n C06∈ 43 σ3 3 We are making two approximations: (1) Replacing nf(z) by 3 z˜ , and likewise for w. Note that σ3 3 4 −1/3 4 by Taylor approximation, nf(z) − 3 z˜ = n O(z ) = n O(˜z ). (2) Replacing (2˜z + 2α − 1) by (2α − 1) and likewise forw ˜. x |x| In order to control the error made, we use the inequality |e − 1| 6 |x|e , so that the error can be bounded by E1 + E2 where Z Z −1/3 4 −1/3 4 |n−1/3O(˜z4)+n−1/3O(w ˜4)| E1 = |n O(˜z ) + n O(w ˜ )|e 1/3 local 1/3 local n C06∈ n C06∈ 3 3 z − w σ z˜3+ σ w˜3−σxz˜−σyw˜ (2˜z + 2α − 1)(2w ˜ + 2α − 1) × e 3 3 d˜zdw ˜ (74) 4zw(z + w) (2α − 1)2 and 3 3 Z Z σ z˜3+ σ w˜3−σxz˜−σyw˜ −1/3 (˜z − w˜)e 3 3 4(˜z +w ˜) E2 = n d˜zdw. ˜ (75) 1/3 local 1/3 local 4˜zw˜(˜z +w ˜) 2α − 1 n C06∈ n C06∈ Notice that in the ﬁrst integral in (74),

−1/3 4 −1/3 4 −1/3 3 −1/3 3 3 3 e|n O(˜z )n O(w ˜ )| = e|n zÕ(˜z )+n wÕ(w ˜ )| < e|O(˜z )+O(w ˜ )|. Hence, choosing smaller than σ3/3, the integrand in (74) has exponential decay along the tails of 1/3 local the contour n C06∈ , so that by dominated convergence, E1 goes to 0 as n goes to infinity. It is also clear that E2 goes to 0 as n goes to infinity. 1/3 local π/3 Finally, let us explain why the contour n C06∈ in (73) can be replaced by C1 . Using the 1/3 local exponential decay of the integrand for z, w in direction ±π/3, we can extend n C06∈ to an infinite contour. Then, using Cauchy’s theorem and again the exponential decay, this infinite contour can π/3 be freely deformed (without crossing any pole) to C1 . We obtain, after a change of variables σz˜ = z and likewise for w, Z Z exp,n 1 z − w z3/3+w3/3−xz−yw I11 (x, y) −−−→ dz dw e . n→∞ (2iπ)2 π/3 π/3 4zw(z + w) C1 C1 exp,n exp,n When adapting the same method for I12 and I22 , we could in principle encounter poles of 1/(2α −1−2z) or 1/(2α −1−2w). However, assuming that α > 1/2, these poles are in the positive real part region, and given the definitions of contours used in Section 4.3, we can always perform the above contour deformations. exp,n Now, we check that the kernel R22 satisfies the hypothesis (3) of Proposition 5.7. Proposition 5.10. Assume that f(x, y) is a smooth function such that all its partial derivatives have exponential decay at infinity. Then, ZZ ZZ exp,n 0 R f(x, y)dxdy − δ (x, y)f(x, y)dxdy −−−→ 0. 22 n→∞

exp,n Proof. It is enough to prove the Proposition replacing R22 by n2sgn(y − x)e|x−y|n. We can write ZZ n2sgn(y − x)e|x−y|nf(x, y)dxdy = ZZ ZZ f(x, y) − f(x, x)n2e|x−y|n − f(y, x) − f(x, x)n2e|x−y|n. y>x y>x 44 We make the change of variables y = x + z/n, and use the Taylor approximations z z2 z3 f(y, x) − f(x, x) = ∂ f(x, x) + ∂2f(x, x) + ∂3f(x, x) + o(z3n−3), n x 2n2 x 6n3 x z z2 z3 f(x, y) − f(x, x) = ∂ f(x, x) + ∂2f(x, x) + ∂3f(x, x) + o(z3n−3). n y 2n2 y 6n3 y The quadratic terms will cancel, so that ZZ Z Z 2 |x−y|n −z n sgn(y − x)e f(x, y)dxdy = ∂yf(x, x) − ∂xf(x, x) dx ze dz+ 1 Z Z ∂3f(x, x) − ∂3f(x, x)dx z3e−zdz + o(n−2). n2 y x so that ZZ Z C n2sgn(y − x)e|x−y|nf(x, y)dxdy = ∂ f(x, x) − ∂ f(x, x)dx + + o(n−2). y x n2 5.3. Conclusion. We have to prove that when α > 1/2, lim H(n, n) < 4n + 24/3n1/3z = PfJ − KGSE . (76) n→∞ P L2(z,∞) Since 4/3 1/3 exp,n P H(n, n) < 4n + 2 n z = Pf[J − K ]L2(z,∞) we are left with checking that the kernel Kexp,n satisfies the assumptions in Proposition 5.7: Hy- pothesis (3) is proved by Proposition 5.10. Hypothesis (2) is proved by Proposition 5.8 (with Bn being Kexp). Note that KGSE(x, y) (defined in Lemma 2.7) does not have exponentail decay in all 2 GSE directions, but since we compute the Fredholm Pfaffian on L (z, ∞), we can replace K (x, y) by 1 GSE x,y>zK (x, y) so that exponential decay holds in all directions. Hypothesis (1) is proved by the following

Lemma 5.11. Assume α > 1/2 and let z ∈ R be ﬁxed. There exist a constant c0 > 0 and for all r, s ∈ Z>0, there exist positive constants Cr,s, n0 such that for n > n0 and x, y > z,

r s exp,n −c0x−c0y ∂x∂yI11 (x, y) < Cr,se . In particular, there exist C > 0 such that

exp,n −c0x−c0y exp,n −c0x−c0y exp,n −c0x−c0y I11 (x, y) < Ce , I12 (x, y) < Ce , I22 (x, y) < Ce .

exp,n Proof. We start with the expression for I11 in (72). We have already seen in the proof of Proposition 5.8 that using both Taylor approximation (70) and Lemma 5.9, Re[nf(z)] is bounded by some constant Cf uniformly in n for z ∈ C06∈. Hence, there exists a constant Cf such that

Z Z (z − w)(2z + 2α − 1)(2w + 2α − 1) 1/3 |(72)| < dz dw e2Cf −n σ(xz+yw). C06∈ C06∈ 4zw(z + w) Now make the change of variablesz ˜ = n1/3z and likewise for w, so that Z Z z˜ − w˜ |(72)| < d˜z dw ˜ (2n−1/3z˜ + 2α − 1)(2n−1/3w˜ + 2α − 1)e2Cf −σ(xz˜+yw˜). 1/3 1/3 n C06∈ n C06∈ 4˜zw˜(˜z +w ˜) 45 1/3 Since the real part of z and w stays above 1/2 along the contour n C06∈, we get that for x, y > 0 |(72)| < C exp − c0x − c0y for some constants c0, C > 0. Moreover, the estimates used in Proposition 5.8 show that for x and y in a compact subset, the kernel is bounded uniformly in n, so that there exists C00 > 0 such that for all x, y > z,

exp,n I11 (x, y) < C00 exp − c0x − c0y . exp,n Applying derivatives in x or y to I11 (x, y) corresponds to multiply the integrand by powers of z and w. Hence, the above proof also shows that for r, s ∈ Z>0, there exists Crs > 0 such that

r s exp,n ∂x∂yI11 (x, y) < Crs exp − c0x − c0y .

6. Asymptotic analysis in other regimes: GUE, GOE, crossovers In this Section, we prove GUE and GOE and Gaussian asympotics and discuss the phase transition. In these regimes, the phenomenon of coalescence of points in the limiting point process – which introduces subtleties in the asymptotic analysis of Section5 – is not present, and the proofs follow a more standard approach.

6.1. Phase transition. When we make α vary in (0, ∞), the fluctuations of H(n, n) obey a phase transition already observed in [6] in the geometric case. Let us explain why such a transition occurs. When α goes to infinity, the weights on the diagonal go to 0. It is then clear that the last passage path to (n, n) will avoid the diagonal. The total passage time will be the passage time from (1, 0) to (n, n − 1), which has the same law as H(n − 1, n − 1) when α = 1. Thus, the fluctuations should be the same when α equal 1 or infinity, and by interpolation, for any α ∈ (1, +∞). The first part of Theorem 1.5 shows that this is in fact the case for α ∈ (1/2, ∞). When α is very small however, we expect that the last passage path will stay close to the diagonal in order to collect most of the large weights on the diagonal. Hence, the last passage time will be the sum of O(n) random variables, and hence fluctuate on the n1/2 scale with Gaussian fluctuations. The critical value of α separating these two very different fluctuation regimes occurs at 1/2. Let us give an argument showing why the critical value is 1/2, using a symmetry of the Pfaffian Schur processes. Lemma 6.1. The half-space last passage time H(n, n) when the weights are E(α) on the diagonal and E(1) away from the diagonal, has the same distribution as the half-space last passage time H˜ (n, n) when the weights are E(α) on the first row and E(1) everywhere else. Proof. Apply Proposition 3.4 to the Pfaffian Schur process with single variable specializations considered in Proposition 3.10, and take q → 1. Let us study where the transition should occur in the latter H˜ model. We have the equality in law ( x ) (d) X H˜ (n, n) = max w1i + H¯ (n − 1, n − x) , (77) x∈ >0 Z i=1 where H¯ (n − 1, n − x) is the half-space last passage time from point (n, n) to point (x, 2). It is independent of the w1i, and has the same law as H(n−1, n−x) when α = 1. For n going to infinity 46 with x = (1 − κ)n, κ ∈ [0, 1], we know from Theorem√ 1.5 (which is independent from the present discussion) that H¯ (n − 1, n − x)/n goes to (1 + κ)2. Hence, ( H˜ (n, n) 1 − κ √ 4 if α ∈ (1/2, 1), lim = max + (1 + κ)2 = (78) n→∞ 1 n κ∈[0,1] α α(1−α) if α 6 1/2. This suggests that the transition happens when α = 1/2. The fact that the maximum in (78) is attained for κ = (α/(1 − α))2 when α < 1/2 even suggests that the fluctuations are Gaussian 1−2α 1/2 Px with variance α2(1−α)2 on the scale n . The fluctuations of i=1 w1i are dominant compared to those of H¯ (n − x, n − 1). We do not attempt to make this argument complete – it would require proving some concentration bounds for H¯ (n − 1, n − x) when x = (1 − κ)n + o(n) – but we provide a computational derivation of the Gaussian asymptotics in Section 6.4. See also [3, 10] for more details on similar probabilistic arguments. In the case α = 1/2, the second part of Theorem 1.4 shows that the fluctuations of H(n, n) are on the scale n1/3 and Tracy-Widom GOE distributed. It suggests that the value of x that maximizes (77) is close to 0 on the scale at most n2/3.

6.2. GOE asymptotics. In this Section, we prove the second part of Theorem 1.4. We consider a scaling of the kernel slightly diﬀerent than what we used in Section 5.2. We set σ = 24/3 as before, r(x, y) as in (67), and here

 2 2/3 exp 1/3 exp  σ n K11 r(x, y) σn K12 r(x, y) Kexp,n(x, y) := .  1/3 exp exp  σn K21 r(x, y) K22 r(x, y) We write Kexp,n(x, y) = Iexp,n(x, y) + Rexp,n(x, y), according to the decomposition made in Section 4.4. The pointwise asymptotics of Proposition 5.8 can be readily adapted when α = 1/2. Proposition 6.2. For α = 1/2, we have Z Z exp,n z − w z3/3+w3/3−xz−yw I11 (x, y) −−−→ dz dw e , n→∞ π/3 π/3 z + w C1 C1 Z Z exp,n w − z z3/3+w3/3−xz−yw I12 (x, y) −−−→ dz dw e , n→∞ π/3 π/3 2w(z + w) C1 C−1/2 Z Z exp,n z − w z3/3+w3/3−xz−yw I22 (x, y) −−−→ dz dw e . n→∞ π/3 π/3 4zw(z + w) C1 C1

exp,n Proof. We adapt the asymptotic analysis from Proposition 5.8. For I11 the arguments are strictly exp,n 1 identical. For I12 , we have a pole for the variable w in 0 (coming from the factor 2α−1−2w in (71)) which must stay on the right of the contour. Hence, we cannot use the contour C06∈ as for exp,n π/3 −1/3 I11 but another modiﬁcation of the contour C0 in a n –neighbourhood of 0 such that the contour passes to the left of 0. We denote by C0∈ this new contour – see Figure9. Instead of working with a formula similar to (72), we start with Z Z exp,n 1/3 z − w 2α − 1 + 2z h 1/3 i I12 (x, y) = σn dz dw exp n(f(z) + f(w)) − n σ(zx + wy) , C0∈ C0∈ 2z(z + w) 2α − 1 − 2w (79) where f is deﬁned in (69). The arguments used to perform the asymptotic analysis of (72) can be π/3 readily adapted to (79). This explains the choice of C−1/2 as the contour for w in I12 in the limit. exp,n For I22 however, thanks to the deformation of contours performed in Section 4.3, the presence of 47 π/3

n−1/3

C0∈

Figure 9. The contour C0∈ used in the end of the proof of Proposition 6.2, for the exp asymptotics of K12 . poles for z and w at 0 has no influence in the computation of the limit, and the asymptotic analysis is very similar as in Proposition 5.8. ¯exp,n ¯exp,n ¯exp,n Proposition 6.3. Assume α = 1/2. Then, we have R11 (x, y) = R12 (x, y) = R21 (x, y) = 0, and for any x, y ∈ R, Z Z ¯exp,n z3/3−zx dz z3/3−zy dz sgn(x − y) R22 (x, y) −−−→ − e + e − . n→∞ π/3 4z π/3 4z 4 C1 C1 ¯exp,n ¯exp,n ¯exp,n ¯exp,n Proof. The fact that R11 (x, y) = R12 (x, y) = R21 (x, y) = 0 is just by definition of R in ¯exp,n Section 4.4. Regarding R22 , when x > y, Z nf(z)−xσn1/3z Z nf(z)−yσn1/3z Z ¯exp,n e e 1 −σn1/3|x−y|z 1 R22 (x; y) = − dz + dz + e dz − , π/3 4z π/3 4z π/3 2z 4 C1/4 C1/4 C1/4 (80) ¯exp where f is defined in the proof of Proposition 5.8, and R22 (i, x; j, y) is antisymmetric. Similar arguments as in Proposition 5.8 show that nf(z)−xσn1/3z Z e Z 3 dz dz −−−→ ez /3−zx , π/3 4z n→∞ π/3 4z C1/4 C1 ¯ so that the Proposition is proved, using the antisymmetry of R22. At this point, we have shown that when α = 1/2, k k exp,n GOE Pf K xi, xj −−−→ Pf K xi, xj . i,j=1 q→1 i,j=1 In order to conclude that the Fredholm Pfaffian likewise has the desired limit, we need a control on the entries of the kernel Kexp,n, in order to apply dominated convergence.

Lemma 6.4. Assume α = 1/2 and let a ∈ R be ﬁxed. There exist positive constants C, c1, m for n > m and x, y > a,

2/3 exp,n −c1x−c1y 1/3 exp,n −c1x exp,n n K11 (x, y) < Ce , n K12 (x, y) < Ce , K22 (x, y) < C.

48 exp Proof. The proof is very similar to that of Lemma 5.11. The contour for w in K12 used in the asymptotic analysis is now C0∈ instead of C06∈, resulting in an absence of decay as y → ∞. The exp,n kernel I22 has exponential decay in x and y using arguments similar with Lemma 5.11. However, exp,n the bound on K22 is only constant, due to the term sgn(x − y).

The bounds from Lemma 6.4 are such that the hypotheses in Lemma 2.5 are satisﬁed, and we conclude, applying dominated convergence in the Pfaﬃan series expansion, that lim H(n, n) < 4n + 24/3n1/3x = PfJ − KGOE . n→∞ P L2(x,∞)

6.3. GUE asymptotics. In this Section, we prove Theorem 1.5. Fix a parameter κ ∈ (0, 1) and assume that m = κn. Again we expect that H(n, m)/n converges to some constant h depending on κ, with ﬂuctuations on the n1/3 scale. exp The kernel K11 can be rewritten as Z Z exp,n z − w K11 r(x, y) = dz dw (2z + 2α − 1)(2w + 2α − 1) π/3 π/3 4zw(z + w) C1/4 C1/4 h 1/3 i × exp n(fκ(z) + fκ(w)) − n σ(xz + yw) , (81) where

fκ(z) = −hz + log(1 + 2z) − κ log(1 − 2z). (82) It is convenient to parametrize the constant κ by a constant θ ∈ (0, 1/2), and set the values of h and σ such that 1 − 2θ 2 4 8 8κ 1/3 κ = , h = , σ = + . 1 + 2θ (1 + 2θ)2 (1 + 2θ)3 (1 − 2θ)3

0 00 With this choice, fκ(θ) = fκ (θ) = 0 and by Taylor expansion, σ3 f (z) = f (θ) + (z − θ)3 + O(z − θ)4. (83) κ κ 3 exp The kernel K22 can be rewritten as  Rexp when α > 1/2,  22 exp exp ˆexp K22 = I22 + R22 when α < 1/2,  ¯exp R22 when α = 1/2, where Z Z exp z − w 1 1 I22 r(x, y) = dz dw π/3 π/3 z + w 2α − 1 − 2z 2α − 1 − 2w C1/4 C1/4 h 1/3 i × exp (n(gκ(z) + gκ(w)) − n σ(xz + yw) , with gκ(z) = −fκ(−z). When α > 1/2,

1/3 Z n(gκ(z)−fκ(z))−|x−y|n σz exp e R22 r(x, y) = −sgn(x − y) 2zdz. π/3 (2α − 1 − 2z)(2α − 1 + 2z) C0 49 When α < 1/2,

1/3 Z n(gκ(z)−fκ(z))−|x−y|n σz ˆexp e R22 r(x, y) = −sgn(x − y) 2zdz π/3 (2α − 1 − 2z)(2α − 1 + 2z) C0 Z 1−2α σn1/3y n g (z)+g 1−2α −xσn1/3z dz − e 2 e ( κ κ( 2 )) π/3 2α − 1 − 2z C0 Z 1−2α σn1/3x n g (z)+g 1−2α −yσn1/3z dz + e 2 e ( κ κ( 2 )) . π/3 2α − 1 − 2z C0 when α = 1/2,

Z 1/3 dz ¯exp n(gκ(z)−fκ(z))−|x−y|n σz R22 r(x, y) = sgn(x − y) e π/3 2z C1/4 Z Z n g (z)+g 1−2α −xσn1/3z dz n g (z)+g 1−2α −yσn1/3z dz sgn(x − y) + e ( κ κ( 2 )) − e ( κ κ( 2 )) − . π/3 2z π/3 2z 4 C1/4 C1/4 0 00 000 000 Since gκ(−θ) = gκ(−θ) = 0, and gκ (−θ) = fκ (θ), Taylor expansion around −θ yields σ3 g (z) = g (−θ) + (z + θ)3 + O(z + θ)4. κ κ 3 Scale the kernel Kexp as  1/3 exp  σn K11 r(x, y) 1/3 exp  1/3 σn K12 r(x, y)  exp,n  2nfκ(θ)−σn (x+y)θ  K (x, y) :=  e  .  1/3 exp  σn K22 r(x, y)  1/3 exp  σn K21 r(x, y) 1/3 e2ngκ(−θ)+σn (x+y)θ

2nfκ(θ) exp By a conjugation of the kernel preserving the Pfaﬃan, the factor e in K11 cancels with 1/3 2ngκ(−θ) exp −σn (xθ+yθ) exp e in K22 (since fκ(θ) = −gκ(−θ)), and the factor e in K11 cancels with −σn1/3(−xθ−yθ) exp e in K22 . This implies that by a change of variables in the Pfaﬃan series expansion, H(n, κn) − hn exp,n P y = Pf J − K 2 . σn1/3 6 L (y,∞) Proposition 6.5. We have that

Z Z 3 3 2 2/3 exp,n n→∞ z − w 2 z + w −xz−yw) σ n K11 (x, y) −−−→ dz dw (2θ + 2α − 1) e 3 3 , π/4 π/4 8θ3 C1 C1 so that in particular Kexp,n(x, y) −−−→ 0. 11 n→∞ Proof. We start with (81). Since for ﬁxed n, the integrand has exponential decay in the direction iφ π/3 π/4 e for any φ ∈ (−π/2, π/2), we are allowed to deform the contour from C1/4 to C1/4 . Then we π/4 π/4 deform the contour from C1/4 to Cθ . This is valid as soon as we do not cross any pole during π/4 the deformation, which is the case when θ ∈ (0, 1/2). The following shows that the contour Cθ is steep-descent.

Lemma 6.6. For any θ ∈ (0, 1/2), the functions t 7→ Re[fκ(θ+t(1+i))] and t 7→ Re[fκ(θ+t(1−i))] d are strictly decreasing for t > 0. Moreover, for t > 1, dt Re[fκ(θ + t(1 + i))] is uniformly bounded away from zero. 50 Proof. Straightforward calculations show that d 1 κ Re[f (θ + t(1 + i))] = −h(θ + t) + log (1 + 2θ + 2t)2 + (2t)2 − log (1 − 2θ − 2t)2 + (2t)2. dt κ 2 2 is strictly negative for t > 0 and θ ∈ (0, 1/2). Calculations are the same for the other branch (in direction 1 − i) of the contour. local Using similar arguments as in Section 5.2, we estimate the integral in two parts. We call Cθ π/4 the intersection between Cθ with a ball of radius centred at θ, and we write π/4 local tail Cθ = Cθ t Cθ . tail Using the steep-descent properties of Lemma 6.6, we can show that the integration over Cθ goes to 0 exponentially fast as n goes to inﬁnity. We are left with ﬁnding the asymptotic behavior of

Z Z (z − w)(2z + 2α − 1)(2w + 2α − 1) 1/3 en(fκ(z)+fκ(w))−n σ(xz+yw)dzdw. local local 4zw(z + w) Cθ Cθ We make the change of variables z = θ +zn ˜ −1/3 and w = θ +wn ˜ −1/3, and ﬁnd

Z Z −1/3 −2/3 n (˜z − w˜) n 3 (2θ + 2α − 1)(2θ + 2α − 1) 1/3 local 1/3 local 8θ n Cθ n Cθ h i hσ3 i exp 2nf(θ) − σn1/3(x + y) exp z˜3 − σ(xz˜ + yw˜) d˜zdw, ˜ 3 where the n−2/3 factor is due to the Jacobian of the change of variables. Finally, we can extend 1/3 local n Cθ to an inﬁnite contour making a negligible error and then freely deform the contour to π/4 C1 . For Kexp,n, the presence of the pole in 1/(2α − 1 − 2z) imposes the condition α > 1/2 − θ, or 22 √ √κ equivalently, α > 1+ κ . √ √κ Proposition 6.7. For α > 1+ κ , we have that

3 3 Z Z z + w −xz−yw 2 2/3 exp,n z − w e 3 3 σ n K22 (x, y) −−−→ dz dw , n→∞ π/4 π/4 −2θ (2α − 1 + 2θ)2 C0 C0 so that in particular Kexp,n(x, y) −−−→ 0. 22 n→∞ π/2 π/2 Proof. Since n > m = κn, all contours can be deformed to the vertical contours C0 or C1/4 in the gκ(z) expressions for I22,R22, Rˆ22 and R¯22. This is because the quantity e has enough decay along vertical contours as long as κ < 1 and n is large enough. By reversing the deformations of contours performed in Section 4.3,

1/3 Z Z n(gκ(z)+gκ(w))−n σ(xz+yw) exp z − w e K22 r(x, y) = dz dw , π/2 π/2 z + w (2α − 1 − 2z)(2α − 1 − 2w) Caz Caw 2α−1 where az, aw are chosen as any value in R<0 when α > 1/2 and any value in ( 2 , 0) when α < 1/2. 2α−1 Since we have assumed that θ ∈ ( 2 , 0), we can freely deform the contours to the contour C|η depicted on Figure 10, where the constant η > 0 can be chosen as small as we want. Now, we estimate the behaviour of Re[gκ] on the contour C|η. 51 C|η

η θ 0

Figure 10. The contour C|η used in the proof of Proposition 6.7.

Lemma 6.8. For any θ ∈ (0, 1/2), the functions t 7→ Re[fκ(θ−t(1+i))] and t 7→ Re[fκ(θ−t(1−i))] d are strictly increasing for t > 0. Moreover, for t > 1, dt Re[fκ(θ − t(1 + i))] is uniformly bounded π/4 away from zero. In particular, the contour C−θ is steep descent for Re[gκ]. Proof. The calculations are analogous to the proof of Lemma 6.6.

Lemma 6.9. For η > 0 small enough, the function Re[gκ(−η+iy)] is decreasing for y ∈ (c(η), +∞) where 0 < c(η) < θ − η, and increasing for y ∈ (−c(η), −∞). Proof. We have 3 2 d (κ − 1) 32y + (1 + 2η) + 8η Re gκ(−η + iy) = , dy 2(1 − 2η)2 + 4y2(1 + 2η)2 + 4y2 from which the result follows readily because κ < 1.

Lemmas 6.8 and 6.9 together imply that for η small enough, the contour C|η is steep descent for the function Re[gκ]. Then, an adaptation of the proof of Proposition 6.5 concludes the proof. exp Turning to the kernel K12 , after the change of variables w = −w˜, we have −1 −1/3 Z Z exp,n σ n z +w ˜ 2α − 1 + 2z K12 (x, y) = dz dw˜ (2iπ)2 π/3 2π/3 2z(z − w˜) 2α − 1 + 2w ˜ C1/4 C−1/4 h 1/3 i × exp n(fκ(z) − fκ(w ˜)) − n σ(z − w˜) , (84) where fκ is defined in (82). √ √κ Proposition 6.10. For α > 1+ κ , exp,n K (x, y) −−−→ KAi(x, y), 12 n→∞ where KAi is the Airy kernel defined in (1). π/4 Proof. We have already seen in Lemma 6.6 that the contour Cθ is steep-descent for Re[fκ(z)], 3π/4 and in Lemma 6.8 that Cθ is steep descent for −Re[fκ(z)]. Due to the term 1/(z −w) in (84), we π/4 3π/4 cannot in principle use simultaneously the contour Cθ for z and Cθ for w. Hence, we deform −1/3 the contours in a n -neighbourhood of θ. Let us denote Cθ6∈ and Dθ6∈ the modified contours, 52 π/4 π/4

n−1/3 n−1/3 θ θ

local local Dθ6∈ Cθ6∈ Dθ6∈ Cθ6∈

local Figure 11. Contours used in the proof of Proposition 5.8. Left: the contours Cθ6∈ local and Dθ6∈ . Right: the contours Cθ6∈ and Dθ6∈.

local local see Figure 11. As in the proof of Proposition 5.8, we call Cθ6∈ and Dθ6∈ the intersection of Cθ6∈ and Dθ6∈ with a ball of radius around θ, as in Section 5.2. Using the same arguments as in Section 5.2, we can ﬁrst deform the contours in (84) to Cθ6∈ and Dθ6∈, and then approximate the local local −1/3 integrals by integrations over Cθ6∈ and Dθ6∈ . By making the change of variables z = θ +zn ˜ and w = θ +wn ˜ −1/3, and approximating the integrand using the Taylor approximation (83) and straightforward pointwise limits, we are left with

3 3 σ z˜3− σ w˜3−xσz˜+yσw˜ Z Z e 3 3 n−2/3 dz˜ dw˜ , π/4 3π/4 n−1/3(˜z − w˜) C1/4 C−1/4 1/3 local 1/3 local as desired. Note that the integration contours should be n C06∈ − θ and n D06∈ − θ , but π/4 3π/4 in the large n limit, we can use C1 and C−1 instead for the same reasons as in Proposition 5.8, so that we recognize the Airy kernel. Let us denote the pointwise limit of Kexp,n by 0 K (x, y) KGUE(x, y) := Ai . −KAi(y, x) 0 The pointwise convergence of the kernel (Propositions 6.5, 6.7 and 6.10) along with the fact that exp,n exp,n exp,n K satisﬁes uniformly the hypotheses of Lemma 2.5 (this is clear for K11 and K22 and it exp,n is proved as in Lemma 6.4 for K12 ), shows by dominated convergence that √ lim H(n, κn) < (1 + κ)2n + σn1/3x = PfJ − KGUE . n→∞ P L2(x,∞) Finally, q PfJ − KGUE = det I + JKGUE L2(x,∞) L2(x,∞) s I 0 K 0 = det − Ai 0 I 0 KAi L2(x,∞) = det I − KAi 2 = FGUE(x). L (x,∞) 53 Remark 6.11. We have assumed in the statement of Theorem 1.5 that m = κn for some ﬁxed κ. 2/3− A stronger statement holds. When m = κn + sn for any s ∈ R and > 0, √ ! H(n, κn) − (1 + κ)2n − 2sn2/3− lim P < x = FGUE(x). n→∞ σn1/3

This change results in an additional factor e2sn2/3−w−sn2/3− log(1−2w) in (81). After the change of variables w = n−1/3w˜ in the proof of Propositions 6.5, 6.7 and 6.10, the additional factor becomes e2n−w˜ and does not contribute to the limit. 6.4. Gaussian asymptotics. In this Section, we prove the third part of Theorem 1.4. Assume 1 1 exp exp that α < 1/2. The factors 2α−1−2w and 2α−1−2z in the expressions of K12 and K22 in Proposition 1.7 prevent us from deforming the contours to go through the critical point at zero as in Section 5.2. We already know that the LLN of H(n, n) will be diﬀerent, and so too will the critical point (its position will coincide with the poles above-mentioned). We have already argued in Section 6.1 that H(n, n) 1 −−−→ =: h(α), n n→∞ α(1 − α) 1/2 1/2 1/2 and we expect Gaussian ﬂuctuations on the n scale. Let rα(x, y) = (nhα+n σαx, nhα+n σαy) exp for some constants hα, σα depending on α. The kernel K11 can be rewritten as Z Z exp z − w K11 rα(x, y) = dz dw (2z + 2α − 1)(2w + 2α − 1) π/3 π/3 4zw(z + w) C1/4 C1/4 h 1/2 i × exp n(fα(z) + fα(w)) − n σ(xz + yw) , (85) where fα(z) = −hαz + log(1 + 2z) − log(1 − 2z). 0 2α−1 0 1−2α 1−2α There are two critical points, fα 2 = fα 2 = 0. Setting θ = 2 , and σ > 0 such that 1 − 2α σ2 = f 00 (θ) = . α α α2(1 − α)2

Taylor expansions of fα around θ and −θ yield σ2 f (z) = f (θ) + α (z − θ)2 + O (z − θ)3 , α α 2 σ2 f (z) = f (−θ) − α (z + θ)2 + O (z + θ)3 . α α 2 Let us rescale the kernel by setting   2 exp σ nK rα(x,y) α 11 1/2 exp  1/2 σαn K12 rα(x, y)  Kexp,n(x, y) :=  e2nfα(θ)−σn (x+y)θ  .  exp  K rα(x,y)  1/2 exp 22  σαn K rα(x, y) 1/2 21 e2nfα(−θ)+σαn (x+y)θ 1/3 We can conjugate the kernel without changing the Pfaffian so that factors exp 2nfα(−θ)−σn (−xθ− h 1/2 i 1/2 yθ) and exp 2nfα(θ) − σn (x + y)θ cancels out each other. Moreover, a factor σαn will be absorbed by the Jacobian of the change of variables in the Fredholm Pfaffian expansion of Kexp,n, so that H(n, κn) − h(α)n y = PfJ − Kexp,n . P 1/2 6 2(y,∞) σαn L 54 exp 1−2α Saddle-point analysis of K11 around the critical point θ = 2 shows Z Z 3 1/2 exp,n z − w z2/2+w3/3−xz−yw σαn K11 (x, y) −−−→ − dz dw zwe , (86) n→∞ π/3 π/3 8θ3 C1 C1 exp Notice that since α < 1/2, θ > 0. Since the contours for K11 in Proposition 1.7 must stay in the positive real part region, we choose the critical point θ. The derivation of (86) follows the same steps as in Proposition 6.5. The most important step is π/4 to find a steep-descent contour. The contour Cθ used in in Proposition 6.5 is not a good choice π/3 here, but it is easy to check by direct computation that Cθ is steep descent for z 7→ Re[fα] for any value of α ∈ (0, 1/2). exp 2α−1 Similarly for K22 , by a saddle-point analysis around the critical point −θ = 2 (the restriction exp on the contours of K22 in Proposition 1.7 impose now to chose the negative critical point), we get that Z Z 1/2 exp,n z − w −z2/2−w2/2−xz−yw σαn K22 (x, y) −−−→ dz dw e . n→∞ π/6 π/6 −4θ2zw C1 C1 π/6 The angle is chosen as π/6 because the contour C−θ is steep descent for the function z 7→ Re[fα], which can be checked again by direct computation. Note that it is less obvious here that we can use vertical contours as in the proof of Proposition 6.7, but it is still possible because the integrand is oscillatory with 1/z decay. exp In the case of K12 , we use the critical point θ for the variable z and −θ for the variable w and obtain Z Z exp,n 1 z z2/2−w2/2−xz−yw K12 (x, y) −−−→ dz dw e . n→∞ π/3 π/6 z + w −w C2 C−1 2 2 exp,n 1 − x +y We recognize the Hermite kernel in the R.H.S. (see [11]), which yields K (x, y) −−−→ √ e 4 . 12 n→∞ 2π Finally, using exponential bounds for the kernel as in Lemmas 5.11 and 6.4, we obtain that H(n, n) − h(α)n G lim P < x = Pf J − K 2 , n→∞ σn1/2 L (x,∞) where KG is defined by the matrix kernel

2 2 G G G G 1 − x +y K (x, y) = K (x, y) = 0,K (x, y) = −K (x, y) = √ e 4 . 11 22 12 21 2π It is clear that KG is of rank one and its Fredholm Pfaﬃan is the cumulative distribution function of the standard Gaussian.

6.5. Convergence of the Symplectic-Unitary transition to the GSE distribution. We have seen in Section5 that when α > 1/2, Kexp is the correlation kernel of a simple point process which converges to a point process where each point has multiplicity two in the limit. The symplectic-unitary kernel KSU is another example of such kernel in the limit η → 0. The framework that we introduced in Section5 – in particular Proposition 5.7 – can be used to prove the following.

SU SU SU Proposition 6.12. Let FSU(x) = Pf J − K 2 where K (x, y) := K (1, x; 1, y) be the L (x,∞) cumulative distribution function of the largest eigenvalue of the symplectic-unitary transition ensemble, depending on a parameter η := η1 ∈ (0, +∞). We have that for all x ∈ R,

FSU(x) −−−→ FGSE(x) and FSU(x) −−−−→ FGUE(x). η→0 η→+∞

55 SU GUE Proof. The limit FSU(x) −−−−→ FGUE(x) is straightforward as K converges pointwise to K η→+∞ and we readily check that we can apply dominated convergence in the Fredholm Pfaﬃan expansion. SU Regarding the convergence FSU(x) −−−→ FGSE(x), the limit of I (x, y) when η → 0 is straight- η→0 forward as well, and we compute that

2 3 2 (y − x) exp (x−y) + 2η − η(x + y) (y − x) exp (x−y) SU 8η 3 8η R22 (x, y) = √ ∼ √ . 4 2πη3/2 η→0 4 2πη3/2 SU 0 Then we readily verify that R22 converges to the distribution δ in the same sense as in Proposition 5.10. Hence, KSU converges as η → 0 to the kernel K∞ in the sense of Proposition 5.7, and we conclude that SU GSE Pf(J − K ) 2(x,∞) −−−→ Pf(J − K ) 2(x,∞). L η→0 L

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J. Baik, University of Michigan, Department of Mathematics, 530 Church Street, Ann Arbor, MI 48109, USA E-mail address: [email protected]

G. Barraquand, Columbia University, Department of Mathematics, 2990 Broadway, New York, NY 10027, USA E-mail address: [email protected]

I. Corwin, Columbia University, Department of Mathematics, 2990 Broadway, New York, NY 10027, USA, and Clay Mathematics Institute, 10 Memorial Blvd. Suite 902, Providence, RI 02903, USA E-mail address: [email protected]

T. Suidan E-mail address: [email protected]